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THE a 


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DIFFERENTIAL AND INTEGRAL 


CALCULUS. 


BY JAMES RYAN, 


AUTHOR OF AN ELEMENTORY TREATISE ON ALGEBRA; THE NEW 


AMERICAN GRAMMAR OF ASTRONOMY, &c. 


NEW-YORK : 


PUBLISHED BY WHITE, GALLAHER AND WHITE, 


W. BE. Dean. Printer. 


1828, 


Southern District of New-York, ss. 
BE IT REMEMBERED, That on the 16th day of July, A. D. 1828, in the 
fifty-third year of the Independence of the United States of America, James 
Ryan, of the said District, hath deposited in this office the title of a Book, the 
right whereof he claims as author, in the words following, to wit: 


The Differential and Integral Calculus, by James Ryan, author of an Ele- 
mentary Treatise on Algebra ; the new American Grammar of Astronomy, §C. 


In conformity to the Act of Congress of the United States, entitled “An Act 
for the encouragement of Learning, by securing the copies of Maps, Charis, 
and Books, to the authors and proprietors of such copies, during the time 
therein mentioned.” And also to an Act, entitled ‘* An Act, supplementary 
to an Act, entitled an Act for the encouragement of Learning, by securing the 
copies of Maps, Charts, and Books, to the authors and proprietors of such cu- 
pies, during the times therein mentioned, and extending the benefits thereof to 
the arts of designing, engraving, and etching historical and other prints.” 

FRED. J. BETTS, 
Clerk of the Southern District of New-York. 


; THE | 
DIFFERENTIAL AND INTEGRAL 


| CALCULUS. 


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TABLE OF CONTENTS. 


oe 
‘PART I 
THE DIFFERENTIAL CALCULUS. 
Gacy. 
I. Preliminary principles. 


IJ. The differentiation of algebraic hiectione of a sin- 
gle variable. 

III. Differentiation of logarithmic and exponontia poi 

tions. . . 

IV, The diftarch eaten a sines nd cosines, and thee 
trigonometrical lines, or the differentiation of cir- 
cular functions. : ° 

V. Successive differentiation. . ; 

VI. Application of the formula of Micladrn aud Tay- 
lor to the development of algebraical and trans- 
cendental functions. : ; . 

VII. Differentiation of equations of po ee 

VIII. Maxima and minima of functions of one variable. . 


. The effect of particular values of the variable upon 


a function, and its differential coefficients. . 


. Application of the differential calculus to the mo 


of plane curves. 


. The determination of singtlac or remarkabts Rotts 


of curve lines. 


. Osculating curves, or the eattntare af ei curve thes : 
. The differentiation of functions of two or more va- 


rlables. , : ie 


. Maxima and minima of functions of ee ra 
. Transcendental curves. 


Pace 


aoe 
Wirt 


“CONTENTS. — 


PART. II. 
THE INTEGRAL CALCULUS. 


Sect. 


I, 


Il. 
HT. 
IV. 

5, 


VI. 


The integration of rational functions involving on- 
ly one variable. 

The integration of irrational finerone: 

The integration of binomial differentials. . ; 

Integration by series. 

The integration of differentials hl eats 
are exponential or logarithmic functions of the 
variable. 

The integration of nfarenians antes aciititine 
are circular functions of the variable 


. Successive integration. 

. The quadrature of curves. 
. The rectification of curves. : 
. The cubature of solids terminated by curve surfaces. _ 
- The integration of differentia! equations of the first 


order and first degree. . 


. The integration of equations of the first ardce apd 


which exceed the first degree. - 
. The integration of differential equations of the se- 
cond order. 


Paces 


181 
208 
217 
229 


238 


ADVERTISEMENT. 


‘Tue inconvenience and great expense of importing Elemen- 
tary works on the Differential and Integral Calculus, induced 
me to undertake the present publication ; and it is deemed un- 
necessary to offer any other apology, at a time when a desire 

of acquiring a knowledge of this useful and important 

Reranch of Analysis has become so - prevalent in the United 
a States. 


The works which I have principally used in preparing this 
treatise, are Lacroix, Lardner, Boucharlat, Garnier, and du 
Bourguet’s Differential and Integral Calculus; Lagrange’s 
Calcul des Functions, Simpson’s Fluxions, Peacock’s Exam- 
ples onthe Differential and Integral iia and Hirsch’s 
Integral Tables. i 


It was originally intended to add an Appendix to this work, 
containing the theory of lines of the second order, and some 
of the general principles of other curve lines: in the course 
of its progress, however, this was not considered requisite, 

| for I understood that a Treatise on Analytical Geometry, 
1 cA embracing also the chief properties of all the remarkable 

curves above the lines of the second order, was ready for the 
press, and would be published in a few months. 


+ , J. R. 
New-York, July 16th, 1828. 


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THE 


DIFFERENTIAL AND INTEGRAL 
CALCULUS. 


ose 


PART, THE FIRST. 


THE DIFFERENTIAL CALCULUS. 


= 


SECTION I. 


PRELIMINARY PRINCIPLES. 


i. In the application of Algebra to the theory of curves, we 
distinguish the quantities which enter into the equation of a curve, 
into constant, such as the axes, the parameters, &c. and variable, 
such as the coordinates ; for instance, in the equation of a para- 
bola, y’=pzx, the parameter, p, is constant, and the coordinates, x 
and y, are variable. A quantity which is supposed always to re- 
iain the same value throughout the whole of any investigation:is said 
to be constant. The name of varzable is applied to any quantity to 
which different values may successively be ascribed. From this it 
is evident, that itis the nature of the question which is proposed that 
determines what quantities ought to be considered as constant, and 
what as variable. Constant quantities are usually represented by the 
initial letters of the alphabet, a, b,c, &c. and variable quantities 
by the final ones, «, y, z, &c. Constant and variable quantities are 
not, however, analogous to known and unknown quantities in com- 
mon Algebra; since a constant quantity may be unknown. 

2. The following may serve as examples of constant and varia: 
ble quantities. A point being given within a circle given in magni- 
tude and position, a line drawn from the given point to the circum- 
ference of the circle is in general a variable quantity as its length 
will change with the point in the circumference to which it is 
drawn. But if the given point within the circle be the centre, the 


same line becomes a constant quantity, being the same length to 
B 


2 THR DIFFERENTIAL CALCULUS. 


whatever point in the circumference it is supposed to be drawii. 
Again, if the base and vertical angle of a plane triangle be given, 
the radius of the inscribed circle, and the distance of its centre 
from the vertex, are variable quantities ; but the radius of the cir- 
cumscribed circle, and the distance of its centre from the vertex, 
are constant quantities ; for the constant ratio which the base bears 
to the sine of the vertical angle is that of the diameter of the ctr- 
cumseribing circle to unity. 

3. Any analytical expression in which one or more variable 
quantities enter, in any manner whatever, either combined or not, 
with constant quantities, is called a function of those variables : 
thus, a-+-bx?-+-c2z° is a function of «; and xy-+-a%x*y?-+c is a func- 
tion ofxandy, Or, ifu = a+-bx?-+cz’, then uw is called a function 
of a ; and, if w — vy--a7z’y’-+-c, then wis a function of x and y. 
Again, if vu = asin. x, we say that u is a function of x, and so on. 

4. As it is necessary to express functions without regard to any 
particular form, a peculiar notation has been invented for this pur- 
pose. The character f, I’, o or, signifies a function; and in order 
to express a function of one variable, one of those, or such like 
characters, is prefixed to the variable, as fx, fy, &c. ; and if in any 
investigation, other functions of the same quantities are to be indi- 
cated, they must be represented by a different symbol, as Fx, Fy, 
&c., observing always to affix, in such instances, the same symbol 
to the same denomination of the function. 

5. If the same symbol be prefixed to two functions of x and y, 
it indicates that they are of the same form, or like functions of x 
and y ; or, which amounts to the same thing, the same operation, 
in any calculation, is to be performed on each of them: thus, fa 
and fy, or Fx and Fy, are two like functions of x and y, which be- 
come identical in making y=. In like manner, f(a--bx) and 
f(a--by) are two like functions of x and y, which become also 
identical in making y = z. 

But if functions of the same form have different constants con- 
nected with them, they cannot in general be represented by the 
same characteristic, in the course of the same calculation. 

In like manner, also, may the function of any two variables, 
« and y, which are independent of each other, be designated under 
the form, . 


S(%s Y), Or O(a, Y) 5 


and so on for all other similar cases. ‘The letter thus prefixed 
must, therefore, not be understood asa factor or coefficient, at 
least in this species of analysis. 


THE DIFFERENTIAL CALCULUS. 3 


6. When an equation between two variables, x and y; is resolv- 
ed, with respect to y, for example, so that we may have 


y = fr, 


y is called an explicit function of x; but if y depends upon x by 
the equation 


S(t, y) = 0, 


then y is called an implicit function of x. In like manner, if u?-+ 
wax--ux-+-1 = 0, uv isan implicit function of 2, being a root of this 
equation. Also, if w = sin. y, and y2?--b2?-+-cx-+-d = 0, u is an 
implicit function of x ; 6, c,and d being supposed to be constant 
quantities. 

The variable x, which is under the sign of the function in y = fz, 
is Called the principal or independent variable, and the other is the 
dependent variable. 

7. The same distinction likewise takes place in functions of ma- 
ny variables, x, y, z, &c. united together by one equation ; this 
equation may or may not be resolved with respect to one of the 
variables, and in the first case, the variables which are under the 
sign of the function are called principal or independent variables ; 
because that on the successive values attributed to these, depend 
those of the variable with respect to which the equation is resolved. 

8. Functions are also divided into algebraic, transcendental, and 
entranscendental. . 

Those in which the variable is united with the constants by com- 
mon algebraical operations, and which may be always expressed in 
a rational form under a finite number of terms, are called algebraic 
functions: such are az’, a+-ba, x”, S/o) &e. ; 

Those which contain exponentials, logarithms, trigonometrical 
lines, circular arcs, &c. are called transcendental functions: as a*, 
x*, alog. x,a sin, x, &c. | 

Finally, intranscendental functious, thus named by Leibniz, are 


those in which the exponentials of the variables are irrational ; as 


2 3 é 
il Fe , Xe. # 


3 
9. Let fx be any function whatever of x, and if x be changed 
into z-+7, the function f(~-++7) may be always developed in a series 
of ascending positive and integral powers of 2, provided that the 
variable x« remains indeterminate, or, which amounts to the same 
thing, that we do not assign any particular value to 2. 
If the function, f(z-++7) be developed, the first term of the deve- 


a 


, 


4 THE DIFFERENTIAL CALCULUS. 


lopment must be fx, because the function not developed must be- 
come fx when? = 0, this function developed must therefore be re- 
duced to fx when 7 = 0; hence it follows that the development 
must contain a term alepcdaest of 7, and that the value of that 
term must be fa. 

In the second place, this development cannot contain any 
term having a negative power of 7, for the supposition 7 = 0 
would render f(x-++7)developed (which then becomes equal to fz,) 


DBL anenee P 
infinite, since it would contain a term of the form a: so that we 


would have fx = ©, or, which is the same, « must have that de- 
terminate value eich renders fx = o, which 1 is contrary to hypo- 
thesis. 


Hence f(«-+7) developed must be of the form 
fet Pit+Pb4+ P+, Xe. 


where the quantities a,b, c, &c, must be integral : forif any of these 
exponents were fractional, there would be as many values of the 
term which involved that power as there were units in the denomi- 
nator of the fractional exponent. Now it is plain, that the radi- 
cals affecting 2 can only arise from radicals included in the primi- 
tive function fx, and that the substitution of x-+-2 for x can neither 
increase nor diminish the number of these radicals, nor change their 
nature, solong as « and zremainindeterminate. On the other hand, 
it appears from the theory of algebraical equations, that every ra-_ 
dical has as many different values as there are units in its exponent, 
and that every irrational function has consequently as many diffe- 
rent values as there are different combinations of the values of 
the radicals which it includes. 


Therefore, if the development of f(z-+7) could contain a term 


™ 


of the form Gz a the function f< must necessarily be irrational, 
and must have consequently a certain number of different va- 
lues, and therefore f(x-+-7) must have the same number; but the 
development of this last in a series being 


fla-bi) = feb PieLP'P+Gi" +, &e. 


each value of fx is successively combined with the values of the 
radical {/7™, so that the function /(~-+z) developed, has more diffe- 
rent values than the same function has when not developed, which 


JHE DIFFERENTIAL CALCULUS. *) 


: 


is absurd. Hence no power of 2 can occur in the development, 
except such as have positive integers as exponents. The series 
must therefore have the form, 


flat) = fet Pi+PP+P?P+, &e. 


‘10. This development of /(2-+:) is therefore true so long as x 
and 7 remain indeterminate, which would not take place if we 
were to give x determinate values ; for it would be possible that 
these values destroy in fx some radicals that would nevertheless 
subsist in f(2-+72). 

Tn order to give an example of this circumstance, let 


fe = Fa-+-"/(a —«), 
Fx being any function of x without the radical ; the function 
Ff (a-+1) shall be F(x--1) X8Y (a--e—a) ; 


and for x = a, a yalue which makes the radical vanish in fz, we 
shall have 


flat) = Flati) xvi; 


so that the development of /f(x-+7), for « = a, would contain 
fractional powers of 7. 
If we still suppose 


fe = f(a¥a), 


. Se . 
in which case 


fer) = Vi (eba) tit = (x-+a)?4+3 Herre) a 
+M(x-+a) 7a Xe. 
we see that if we can make « =—a, which Aypaests makes the 
radical disappear in fx, the coefficients of 7, 72, &c. become infi- 
nite, and at the same time the function f(x-+7) becomes 4/2. 


These sorts of cases, and the consequences resulting from them, 
snall be examined in a subsequent part of the work. 


11. The development of f(x-++2) may Bs put under any one 
whatever of these forms 


S(a+t) = fatpe 
= fat Pt +92” 
= fat PitP?+ri’, 
+c, 


6 THE DIFFERENTIAL CALCULUS. 


by putting My 
’ p = P+Pit+P"@+, &e. 
qg= P+ P+ P"+, & 
ros PP +P 77+, &e. 
&ce. 

And thus it becomes easy to demonstrate that any term in the 
series representing the development of f(a-+7), may be made 
greater than the sum of all the terms that follow, a theorem that is 
of great use in the application of this calculus to the geometry of 
curves. . 

In fact, 7 not being in fx, nor in the functions, P, P’, P’, &c, we 
we may always conceive 2 as small as we please. 


1. fx >pt, and consequently 
fe>pitP'?+P?+,&c. ; 
2. P>q, hence Pi >q’, that is to say, 
Pi>P'?+P#+,&e. : 
3. P’>rt, hence P'?>772°, that is to say, 
P?> P+ PP" +-,&e. ; and so on. 


12. If uw = fx, and if x becomes x-+2, the function wu changes 
its value and becomes w’; we have therefore u’ = f(x-+7). Now, 
if 7, the increment of x, be supposed successively to assume all 
values from zero to infinity, the function f(z-+z) or w assumes a 
succession of corresponding values ; that is, uw’ becomes wu”, and x’ 
assumes another new value uw", and so on for every corresponding 
value of 7. The ratio of the variation of the function u compared 
with the increment of the variable x, upon which the value of this 
function depends, is the immediate object of the differential Calcu- 
lus. There are three methods of presenting this calculus to the 
student, all of which arrive at the same end, though by different. 
means. 

13. Newron first proposed to determine the ratio of the velo- 
cities with which the function and variable increase or decrease, 
and called these velocities their fuxions ; the function w and its va- 
riable « being called fluents. The notation by which he expressed 
the fluxions is u, z,and consequently the ratio of the fluxions is 
expressed thus, 


ub 


ce 
This is called the Auxzonal coeffictent. 


THE DIFFERENTIAL CALCULUS. 7 


it may be observed here that fluxions are quantities absolutely 
indeterminate, but may have any values, provided that their ratio is 
that of the velocities with which the function and variable change. 
The fluxtonal coefficient, however, is given for any particular value 
of x, and in general only varies witha. 

According to this method, quantities are considered to be gene- 
rated by motion, as lines are produced by the motion of a point, 
surfaces by that of a line, &c. The fluxional calculus is therefore 
a method of determining the velocity with which a function varies 
at any point of time compared with the velocity with which its va- 
riable changes. 

14. The conception of motion and time, which are involved in - 
this method, were considered inconsistent with the rigour of mathe- 
matical reasoning, and wholly foreign to that science. 

As animprovement upon the principle, D’ALEmBerrt proposed the 
method of limits. Considering wu as a function of 2, let the varia- 
ble x bé supposed to receive any finite increment 2, so that it be- 
comes x-+7, and: let the corresponding value of « be w’ so that we 
shall have the equations 


>, = fa, 
eeu = f(t); 


ie 
and let the value of —- 


be found: this will bein general a quan- 


tity whose value will ivan on those of z and 7, and it will express 
the ratio of the finite increment wv —vw of the function, to 7 that of 


the variable. If in this quantity 2 be supposed to be = 0, it will 


express the limit of the ratio of the corresponding variations of the 


function and variable, these variations being reduced to infinite mi- 
nuteness, or being considered as continually approaching to zero. 
15. Hence the method of D’ALemBERT is nothing else than 
to determine the value of the fraction having for its numerator and 
denominator the simultaneous increments of the function and va- 
riable, when both these increments are = 0, or the fraction itself 
becomes x The value thus determined is called the differential 


coefficient ; for instance, let 


TEER MeN chie es, whe oo. 0 Vai ths ol step aD) 


if « becomes x-+-2, u becomes w’, and we have 
ul = (x-+72)’, 
and by performing the operation indicated, 


THE DIFFERENTIAL CALCULUS. 
w = +30 307Ee 5 
subtracting from this equation, the equation (1), there will remain 

U—u == 3127-+-37722-+-2°, 
and dividing by 7, we have 


U =u 
ne SRP 4b Sig ate eh IE ae (2) 


Now, uv —w represents the increment of the function w, and 7 that 
UW, ; 4 
Tis therefore the ratio of the increment of u to 


of the variable x : 


that of the variable x. By considering the second member of the 
equation (2), it appears evident that the ratio diminishes according 
as 2 diminishes, and that when 7 becomes nothing the ratio is re- 
duced to 32”. i 

Uw—i 


This term 3z? is therefore the limit of the ratio—— < it 1s to- 


wards this term that the ratio approaches according as 7 diminishes, 
and to which it may approach as near as we would wish. 
16. In the hypothesis of : = 0, the increment of u becomes also — 

: u— 0 : 
nothing, ; “ reduces to i? and consequently the equation (2), 


becomes 


‘This equation presents no absurdity, because algebra shows that 
may represent all sorts of quantities, provided it arises from 
operations previously performed upon algebraical quantities. 

Moreover we conceive, that since in dividing the two terms of a 
fraction by the same quantity, the fraction does not at all change its 
value ; it follows therefore that the minuteness of the terms of the 
fraction has no influence upon its value, and that consequently it 
may remain the same when these terms have arrrived at the last 
degree of minuteness, that is to say, when both become actually 
equal. to nothing. 


17. The fraction > is a symbol which has replaced the ratio of 


the increment of the function to’that of the variable: as this sym- 
bol leaves no trace whatever of that variable;two. indeterminate 


THE DIFFERENTIAL CALCULUS. q 


quantities, du and dx are assumed, so that we shall have in equa- 
tion (3) 


In this equation, du is called the differential’ of the function, and 
dx that of the variable. The notation du, dx, is not meant to ex- 
press d Xu, d Xz, but simply “ the differential of u” and the differen- 

du 


tial of «x. aa? rather its value 3x”, iscalled the differential coeffi- 


sees du, . 
cient of the function ~: it is proper to observe that — being the 


sign which represents the limit 32° [as equation (4) shows] dx 
should be always placed under du. Notwithstanding, in order to 
facilitate the operations of Algebra, the denominator is sometimes 
taken away, and equation (4) becomes in that case 


du = 32°dx. 


This expression, 32"dz, is called the differential of the function wu. 
It is evident that the differentials of the function and variable, 
according to this sysiem, are the same quantities as the fuztons in 
the Newtonian method, differing only in notation and name. 

18. The third and last method is that of Lacrance. He equai- 
ly rejects the limits of the ratios of D’Alembert, and the motions 
and velocities of Newton, and has proposed fundamental principles 
for the calculus at once rigidly demonstrable and purely analytical. 
Let uw = fx and uv’ = f(x-+7), the function uw’, or its equal f(x+7) 
may, in general, (as has already been shown) be developed in a se- 
ries of this form 


flati) = fr-+-Pi+PP+ P-L, &e. 


or 


w = ut Pit+PP?+P'P+, &e. ; 


in which P, P’, P’, &c. shall be new functions of x derived from 
the primitive function fx or u. Although the form of these de- 
rived functions depends essentially on that of the primitive func- 
tion, still there exists among them a general law, as will be shown 
in a subsequent part of this work. 
The function P, is called by Lacrance the first derived function 
of the primitive function u, and the notation he uses to mark this 
new function of x,is f’x. This may be shown to be the same 
quantity which D’ALemspent calls the limit of the ratio of the cor- 
C 


10 THE DIFFERENTIAL CALCULUS. 


responding increments of the function and variable : for let 4 be 
transposed in the preceding equation, and both members be divided 
by 2, and we shall have | 


/ 
U 


“ae = P+Pi+P'?+, &e. 


If 2 = 0, the second member becomes P, which is therefore the 
limit of the ratio. 

19, From what has been already observed, it appears that this 
is the same as the fluzional coefficient of Newton, and the limit of 
the ratio or differential coeficient of D’Avemsert. It is also evi- 
dent that the second term of the series is the oi ae of the 
function u, z being assumed as dz. 

' The notation of the differential calculus shall be adopted i in the 
following treatise, because it is. that which is used in most of 
the modern works written on this subject. 

The process by which the differential coefficient of a function is 
found, is called differentiation. 


SECTION II. 
The Differentiation of Algebraic Functions of a Single Variable, 


19. We shall in this section explain the methods of differentiat- 
ing those functions which occur in the Elements of Algebra, such as 
the sums, differences, products, quotients, powers, and roots of al- 
gebraic quantities. 


In the first place, it is very important to remark, that in the equa- 
tion 


Bee a ea ue Ne ee ene 


where fx is an algebraic function composed in any manner whate- 
ver of the single variable <, ifwe put x-+, or, which amounts to the 
same thing, «-+dz in the place of x, and then by developing f(a-- — 
dx), by the known rules of common Algebra, according to the po- 


sitive and integral powers of dx, we shall obtain (Art. 9) a series of 
the following form. 


flatde) or uv =u+-Pdz-+P'(dr)?+-P'(da)t+-, &e.. . . » (2) 


THE DIFFERENTIAL CALCULUS. ii 
which may be also expressed (Art. If) thus : 


wu =u+Pdz+p(de) ; 
where p is equal to P’-++-P"(dz)4-P" (dx)?-+, &c. 


Now, subtracting equation (1) from (2), or, which is the same 


thing, transposing the first term of the second member of equation 
(2), and we shall have 


w—u = Pdr-+-P'(dx)?+-P'(dz)-+-, &c. 
‘therefore, by division, 


Pp 


UW mm 


~~ = P-tpde, 


and, by taking the limit, we shall have | 


dz da 


Hence it follows that the differential coefficuent is equal to the co- 
efficient of the term which contains the first power of da in the de- 
velopment of f(x-+-dx) ordered according to the ascending powers of 
dx ; and the differential ts the second term of that development. 

This furnishes the means of generalizing the process of diffe- 
rentiation : and we shall now proceed to illustrate it more fully by 


its application in finding the differentials of a variety of algebraic 
functions. 


20. Let fx be x-+-2", or, which amounts to the same thing, let 
u = 2a? ; 
then, putting x-+-dz for x, we shall have . 
oe a atde-+(2+-de)?; 


or, by developing and arranging the terms according to the powers 
of dz, 


a = eta? (14-20)de+(de)? ; 
ie by transposing and substituting w for x-}-x*, we shall have 
. w—u = (1-++-2x)de-+ (dx) ; 
Se ae by dx, and eng the limit, 


W <m t du 
a Ab 22, or = 1-}-2w. 


12 THE DIFFERENTIAL CALOUDUS. 
Again, let fz be a-a--bx; then te 
; uw tha+be > 

and putting «--dx for x, we shall have 

uo = ta+b(r+de) ; 

Ha Maeda le ‘ai 

or, vw = ta+batbdz ; me . 
al tf me 1b — bdz, : 


du 


and ar 


=ilbs so du == baa. 
fies» - 

Hence it follows, that constant quantities combined with a fune- 
tion by addition or subtraction, disappear in tts differential, and alt 
constant quantities which are combined with it as factors are simi- 
larly combined with its differential. _ 

The same observation obviously applies to constant divisors, 


1 
since b may be x 


21. Let us now proceed to find the differential of the function 


é rv 


u = fa = ap-+bg-+er+, Ree 


Ps q, 7, &c. being algebraic functions of a, anda, 6, c, &c. con- 
stant. Putting x-+4-dz for «x in each of the functions, p, 9, 7, &c. : 
thus ) , | 


eo 


p becomes p-+p'd«-+S(dax)?, where S = p"--p"dzx-+, &c. 

gq becomes q-+-q'dx-++-S'(dx)’, where S'= q"-+-q"dx-+, &c. 

¢ becomes r-+r'dx-+S"(dx)’, where S" = r’-+-r"dat+, &e. 
i Xe. 


So that we shall have NG ay . 


flatda) = ap+-bq-Ler+, &e. 
-+-(ap'+bhq'+er'+, &e.)dx 
+-(aS-+-bS +c8’+, &e.) (dz). 


Transposing and substituting w’ for f(x++-dx) and u for ap-++bg-+- 
cr-+-, &c., we shall have 


uu = (ap'--bg er’ +, &e,)da-+ (aS-+-bS'+-c8""+-,&c.) (dx)? ; 
:, dividing by dx, and taking the limit 4 


d t n ’ 
- = ap'-+bq/ ber’, &e. 


cw 
Aa f 


ty SHE DIFFERENTIAL CALCULUS. 13 


s 


or du = ap'dx-+-bq'da--cr’ dx-+, &c. ; 
but pdx, q'dx, r'dx, &c. are the differentials of the functions, p,9,r,&c. 
we shall therefore have 


‘du = adp-+t-bdq-+cdr+-, &c. . ... 6. (1) 


Hence it follows, that the differential of a function which is com- 
posed of several funetious of the same variable united by addition 
or subtraction is in general the differentials of the several functions 
united together in the same manner, and with the same signs as the 
; tapi themselves. 


These functions, p, 9, 7; &e. may be any BZcUralc functions 
whatever of the variable « ; for instance, let. “ 


u = a(e-ba'n*) -+6(8'x*++c'2°) Pele aett-d)+, So. 


then, by substituting ¢-+de for x in each of these expressions, and 
proceeding as in Art. 20, we shall find dp = dz-+-2a'xdx, dq = 2b’ 
xdx+-3c'27dx, dr = 4dzx —82%dx, Xe. 


*, du = a(dz-+2a'adx)-+-b(2b'xda-+3e'x*dx) --¢(4da—825dx)-b, 
fen Ae &e. ’ 


or du = (a-+-2aa'm)da-+(260'x-+Sbe'n*)de+ (Aci -8cx*)\dx+, &c. 


22. To differentiate a function which is the product of two or 
more functions of the same variable : let, in the first place, 


uw = fr = = apa, 


p and q aah any algebraic functions of x, and a a constant quanti- 
ye substi? x-+-dx for x, p will change into 


pty saiaelh 
and q will become 

qtqdx-+-S'(da)* ; 
the function proposed becomes therefore 


fla-t-de) = apq-+(agp'-+apq)de-tap'g (dx), &c. 
Transposing and dividing by dz, we have 


d ’ , ee 
re = agp +apq'--ap'qdx--, &c. 
and taking the limit, by making dx = 0, 


du ; Pas 
i Pie agp’ --apq ; 


14 THE DIFFERENTIAL CALCULUS. 


°, du = d(apq) = agp'du-+-apq' dz. 
Now p’dz and q‘dz are still the differentials of p and q ; therefore, 


du or d(apq) = aqdp--apdgq. 


Hence the differential of the product of two functions of the same 
variable, is obtained by multiplying each function by the differential 
of the other, and adding together the two results. « 

Thus, if u = (2° —2a*) X (a? -3a?) ; then, we shall have 


du = (2? —2a?) X d(a? ~— 3a?) -+- (x? — 3a”) X d(x? —2a”) ; 
but d(2?—3a) = 2xdzx, and d(x?—2a?) = Qxdx ; 
“du = (x?— 20?) 2adx-{- (17 = Bu?) Qada, 
or du = (427— 10a2x) dx. 


We shall find in like manner that the differential of the product 
pq’, pgr being functions of the same variable, is pgdr--prdg+ 
qrdp ; and, in general, if « be equal to the product of any num- 


ber of factors, p, g, r, &c. which are functions of the same variable 
x, we shall have ) 


‘ 


du = qrs...dp+-prs...dq+pqs...dr-+pqr..ds+, &c. , 


Hence it is evident, that whatever may be the number of factors, 
the differential of the product will always be equal to the sum of 
the products of the differential of each multiplied by all the others. 

23. The rule for differentiating the function 2", when m is a 
whole positive number, may be readily deduced from the preceding : 
in fact, if we put x™ under the form of a product, we shall have 


U=—Fe Le ELL 


m—t m—i m—1 
~du=a data dzxtx dzx+, &e. 
As the number of terms in the second member of the equation, 


" m—1i , 
are m, and each equal tox dz, their sum shall be ip 
m—t1 ie 
du=mx dz. 
24. The differential ofa fraction whose numerator and denominator 
are each functions of the same variable, may be obtained: thus, let 
w= fe = Lf, 


g 


THE DIFFERENTIAL CALCULUS. | 15 


p, and g being functions of the variable x; then, uy = p, and from 


what precedes, we have | 
) dp = = udtrede : 
- du= deh, 8 
* 


and by substituting in the second member for u its value 7 we shall 


Air by 
have * 


dp pq 

AE Sat ee 
qe 
or, reducing the fraction to the same denominator, 


_ 94p- a 


hence this rule for seed this eee of a fraction: multiply 
the denominator by the differential of the numerator, subtract from 
this product that of the numerator. multiplied by the differential of 
the denominator, and divide the difference by the square of the de- 
nominator. 

The differential of a fraction may be likewise found in the fol- 
lowing manner : putting ; a for x, we shall have 


fleseile) = PAPC As)* 


a 


q+q dx-+-S"(dx)? ” 


The division of the numerator by the denominator, shall give 
this quotient — 


fla+-de) = ul (( ir )dx-+Q(dx)?+-, &e. 
: 2 ‘dx— ‘dx 
du = Pure dz = ee OP 
( ra q 


Now, pdx and q da are the differentials of p and q; we have 
therefore 


du or a’) = gap — wel aas ip 


the same as before. 


25. We have already seen in Art. 23, that the differential of the 


m—1 


m 
function: « is ma dx, whenmis a whole positive number, we 


if 


= 


16 THE DIFFERENTIAL CALCULUS. 


m oe 
shall now show that the differentials of the functions a",andz ”; 


wh i 
in lg m 


- am m 
are —-a” dx,and ——x ” dz, 
11 17 
For this purpose, let 


usa 5 Ut a™ 5 


thus putting successively, z = um, z = 2%, 
the numbers 7 and a being supposed integral and positive, we have 
from the article above alluded to, 


dzics Ode, dz = ma™—dx 
2. nu du = mz™—' dx, 


and consequently, 


nm yr—} 


m 
n an em} 
du or d(x ) ta, dx. 


\8 


Putting for w*—! its value x” =, we shall have 


m cm—t m ——} 
d(z *) — ae dx= —. unr dx. 
nm 


1) 


Let, in the second place, u = 2 “n; whence u"=a—"; and put- 


; j 
ting z = uv", and z = a= On we shall have (Art. 23,) 
dz = nu"—!du ; 
MT fhe ; : : 
then by comparing = with the fraction > (art. 24.) which gives p 


—1i 
= 1, and g= 2”, we find dz = — bad 


dx = — ma—™—'dx ; 
£ 7m fh 
2. nu"—ldu FF id a a 


* See my Elementary Treatise on Algebra, Art. 86. 


THE DIFFERENTIAL CALCULUS. 17 


2 | 
Substituting for w—1its value «<*>, we shall have 


Hence this rule for differentiating any power whatever of a va- 
riable quantity : multiply the differential of the variable by its ex- 
ponent, and the product by that power of the same variabie whose 
exponent rs less by unity than the given expunent. , 

This rule might have been immediately deduced from the deve- 
lopment of the binomial (x-+-dx)™, since (x-++-dx)"= «™-+-ma—* 
dx-++-, &c. whatever may be the value of m, (Alg. 450.) and con- 
seunenty, (Art. 19.) the second term ma™—'dz is the differential of 


= 


26. We can apply the preceding rule to the differentiation of 
functions of the form (a--bx™)" : for, let a4-ba"= z ; then 2"= 
(a+bx™)", and 2*—1= (a+-bz™)"—!. By taking the differentials, 
we shall have, ; 


dz = mbx™—"'dr, and d(z") == nz™—!dz5 


-, by substituting for z*—1 and dz their values, in the last expres- 
sion, we shall find 


d(2") = n(a--ba™)"—! X mba" dz ; 
*, if u = (a+b), we have 
du = nmba™—! X (a--ba™)"—! da, 
We shall find, in the same manner, that the differential of (a-+-2)" 
is Wr ee ; that of (a+627)°, 3(a-+-b2")? X 2brdz ; and that 
of (a2-+-22)? is 3(a?--2? 2 X2adz. 

If we had u = (a--br-tex?)?, we should consider the trinomial 
a+bz-+cx’, as a particular function z ; then 2 (a + be-+ex?)?, 
and 27? = (apbe-beat) 

Now, d(2?) ee 2 Xdz, and dz = bdx-+-2cxdz ; 


.. by substitution, we have 


du = d(z?) = Ma a+ ba-ben?)- 2X (bde-+2eadn), 
D 


18 THE. DIFFERENTIAL CALCULUS. 
dz bdz+-2cadx 
and du = ——~- = cm 


2/7) o(aeborten®)? 


As we have frequent, occasion to differentiate radicals of the se- 
cond degree, it is wel! to notice, that according to the formula 
dz 
2/2” 
by dividing that of the quantity contained under the sign by double 
the radical itself. 
27. If u = fy and y = ox, from what precedes, we may readily 
determine the differential coefficient of u considered as an implicit 
function of «3 we can likewise obtain the differential coefficient 


» the differential of a radical of the second degree is obtained » 


, by eliminating y bet ees these two equations, before we apply 


the process of differentiation ; but without having recourse to that 
preliminary operation, we can determine immediately the difle- 


rential coefficient, in the following manner. 


dx 
Let y = o(a-+71) = gr-+Pi+S’, 


and u’ == f(y-++k) = fy PRESE 
where kis supposed ==7'—-y or Pi+ S52. ; ‘ 


Qi len Y=! — P+Si, and “ ims = Pk. 


Multiplying the corresponding members of these two equations 
and we shall have 


lite: 


(P-+Si):(P'-ES'k) ; 


when? = 0, k also vanishes, since it has been put = Pz-4-S:’, we 
dy du , dy du _ . 
have therefore ee P, — ag =P, aut * == PP’) 


and consequently 


Hence, it follows that, if there be three quantities u, y, x, each a 
function of the other, the differential coefficient of u considered as a 
function of x is equal to the product of the differentral coefficients of 
u considered as a function of y, and of y considered as a function 


of 2. 


THE DIFFERENTIAL CALCULUS. is 


it is obvious that by continuing the process, thé same principle 
may be shown to be applicable to any number of differential coef- 
ficients. This method is of great utility in facilitating the diffe- 
rentiation of complicated ae for instance, if u = ene 


(b=) Hs; OF fi V(b- 3) 3 then 


€ 
u = fy = (a-+y)', and y = px = /(b—-—.); 


*. by differentiating thesé two equations and taking the differen- 
tial coefficients, we shall find 


du ' pe k 
a ty = Ahatyb—3)} 
\ dy _ c 
dx c. 
ave Te) 
a= ‘fot y(o— = Sai) 33x -——, 
03 of (b— > 
and consequently - 


in Mage) 


ae | Soe ee 


“  2y(b=5) 


Examples in the differentiation of algebraic functions of one 
variable. 


a OS ee ee ae 
28. Ex. 1. Ifu= /(Qre—at) 2. du igen 


Px, 2. if usa(l+r)} . du = (1-+-22)dz. 
Ex. 3. Let u = 2°(a+-2)*. 
du = (36+ 52).(a-x)0"dx. 
Ex. 4. Letu = (abe). (b-+-2)”. 
du = {nb-tma} (n-bm)e (aba)! Cuan Foyt 
Ex, 5. Lefu = (1 pey(teys” | 
b= $4462 100?? (1 x)t1-+-xr) dx. 


ois TH taj “AL, 


20 THE ee eee CALCULUS. 
Ex. 6. Letu = a(1--a) (1px)? 
du = §1--2o-FQat-+ 49°} dr. 
Ex. 7. Let u = (a+2)(b+2)(c-+2). 
du = }ab--ac-+be+2(a-+b-+-c)x+-327} dx. 


Ex. 8. Let u = “. 


a 
Ex, 9, Let u = eae 


__ sada 
(Gx) 


Ex. 10. Let vu = 


4adx ~ 


~ (2)? 


a” “Te 
Ex. 11. Let u= (i-Fay" % 
net de 
(eet 
ier 
a2 --ael 
ch 2(a*—~1)dx 
(x®--a- 1)? 


du 


du 


Ex. 12. Letu = 


du 


Ex. 13, Let uv = —_. 
: He 


Ex. 14. Let vu = (1--a),./(1—2). 
| __ (1—32)dz 
2/1 ay 


a es ee 


2f(l—2).f (l-ba’y 


THE DIFFERENTIAL CALCULUS. 9} 


Ex, 16. Letu = Jf (+24) + 2’) 


VU+#)- V2) 
du _ Siete fe 2 | 
dx meat ime Ge eee 
Cc e 
Ex. 17. Let u = 0+ “lems w/e at 
du 2b Ac Qe 
., See Sayyae * 3223/x x3 
Ex. 18. Let u = 0 Sok Boe Dy 


(e+e)! 
du _ — {323-f2? —60—4-+2(1 782) /(1— das a 
ae BA aati Het (apa) 5 
Ex. 19. Let u = 


eV 
du.  1-F2z? 
dx (Ix?) 
; _ 5, a vu? 
Ex. 20. Letu = ae ety av fi ae 
1] 1 
Sits ee ee 
du Siu ral x)? nw) 
RO OF, OTT 
| e | y(t—2) 
S ; ior J (1—a) . x ¢* 


SECTION Ill. 


Differentiation of Exponential and Logarithmic functions. 


29. The function x”, in which z is the variable, and mis con- 
stant, leads to the consideration of the function a*, in which the 
exponent is the variable, and a constant. Functions, having va- 
riable exponents, are called exponential functions. 


22 rf THE DIFFERENTIAL CALCULUS. 


In order to find the differential of the function a*, let x-+2 be 
substituted for x, this substitution gives the functions 


w = art =a" Xa‘. 
Brite a = 1--4, we shall have, 
a’ == (t+-6)' ; 
this being expanded by the binomial theorem gives 
= (148) = 14 CED ep MOE hy, we. 
Arranging the terms of this series according to the powers of 7, 


it is evident that the first two terms of the development of «’, 
shall be | 


62 | 6 bt 
tx SE Sp BOE i A ar Na. 
Hb- Ste — T+, &e)i 


Let the remaining terms be represented by Si”, and for brevity, let 


h2 ~ hs b4 
eo Star aac 
| a — ee a—1)* a—1)* 
= (a— Tl) ~ Sa. «dank nar Con &e. 


thus we shall have 
at = (1--e)e-+ Se? ; 


multiplying both members by a? and substituting wu for a*, and wu for 
a*a', the result after transposing wu, shall be 
w—u = catt-+Sa7r?. 
Hence, by taking the limit, we have 
d 
oh scot, du = catde. 
dx 
‘The coefficient c depends, as we have seen above, on the quantity 
a, which is as the base of the exponential, and this. coefficient is 
usually called the modulus. Hence it follows, that the differential 
coefficient of an exponential quantity is equal to that exponential 
multiplied by a constant coefficient, whose value depends om the base 
of the exponential, 
30: Itis:evident from what has been observed‘ (Alg. SOT.) that 
any variable quantity may be expressed’ by a constant raiyed’ to a 


LHE DIPPERENTIAL CALCULUS. 23 
if 

variable power ; then the exponent of that power becomes a func- 
tion of the same quantity, and that function is called the logarithm 
of the proposed quantity : thus, if vw = a*, a is the logarithm of u ; 
and if log. x denotes the logarithm of x, we shall have this equa- 
tion, a = a’’s-*, in which a is the base of the logarithmic system. 
And, if lx denotes the logarithm of x in the Neperian system, in 
which the base ise,* the general formula « = a’s-*, will become for 
these logarithms, « = e'* ; so that, in general, we shall have 


‘y — 1 > 
qlog.2 —= elt ; 


in order to find the differential coefficient of the logarithm of a 
variable quantity, we shall assume, in the equation u = a’, log. 
u = 2, relatively to the base a ; 


.. d. log. u = dex. 
Kliminating dx by this, and the equation 
du = ca*dx, 
found in the last article, we shall find, by putting 4 = a", 
d. log u 1 1 du 


et 5 ai d. log wee A, 


du cu aa Guts 


d. log u 1 “3 au 
lab is = M.—' 3 WF . d. log RO en IT ba 
du u u 


Hence, there results this general rule: that the differential coefji- 
cient of the logarithm of a variable quantity is equal to the product 
of the reciprocal of that variable multiplied by a constant quautity 
whose value depends on the base of the system of logurithms. 
31. For the Naperian system, the modulus being equal to unity, 
we shall have 
CAT ee | _ du 


=-; .dlu=—. 


so that, i this system of logarithms, the differential coefficient of 
the logarithm of a variable quantity is equal to unity divided by 
that variable. Frequent use is made in calculations of the expo- 
nential quantity e”, e being the number whose logarithm is unity 

The differential of this quantity, from what has been just laid down, 
is readily found to be etdx. For, let w = e* ; taking the logarithms, 


* The value of the Naperian. base will be determined hereafter (55). 


Co 


D4 THE DIFFERENTIAL CALCULUS. 


according to the Naperian system, we shall have 


d : 
begs "leg iines — = leda; and since le = 1, we have = ge 
*, du = d. (e”) = uda = eda. 


32. We will now give some examples of the application of the 
rules for the differentiation of logarithmic functions ; and for the 


sake of simplicity we shall always suppose them Naperian loga- 
rithms.* iy 


Ex. 1. Let u = {ore , Cz a ae making —_—; Fat 5 <p s 


we have du = ad ‘+: but dz = nts. 


F du dz ip 1 a? 
"* dzdx 2" 


(a2 Pa")? 
2 2 
consotueny mee “- ae Ss5F 
(a?-+2") 
we a? 


and 2 = a(a?--2?) 


Ex. 2, Let u = 2x ; that is to say, the logarithm of the loga- 


rithm of x. 
Ae ie 
Ae 


Put lx =P we have u = Iz; 


is #4 
ee du = —, and dz = —-. 1B: 
du dz hal 
Hence, cn oe 
du 1 
d = ae! 
an Sars Weds dite 


Ex. 3, Let v= “Pr, where Ba, is ‘ie log of the Wee: of the log 
of x, orl(Px). 
By taking the differential, we have 


* The method of determin’ ng the constant quantity, usually called the modulus , 
by which we multiply a Naperian logarithm in order to pass to the corresponding lo- 
garithm of another system, shall be explained in a subsequent section. 


“e 


THE DIFFERENTIAL GALCULUS. 95 


d _ alr 1 dr 
OTe Te ale 
_ du I 
**dx lal?” 
1 , 1 
Ex. 4. Let u = l=.» Since = 11— lx; 
x liad 
dx 
we have du = d.(I1—lx) = — rye 


Ex. 5. Let u = (pq) ; where p and q are both functions of «. 


Since l(pq) = lp+-lq, we have 
du = d.(lp+lq) = d.lp-+-d.lq : 


re dae os 89: 
r.. & 


If dp = pdx, and dg = q'dz ; then, 


Ex. 5.. Let «= CES. 


Since 1 e = l(a-+x)—Ka-~x); we have 


du =d.l(a-+-x) —d.l(a—~—x) ; 
dx dz 


Gane ajt-% aa 
d yea ee 
and consequently, — = ———3° 
. eae ts amr 
Ex. 6. Letlu = 1} 4 i 
V(i-ta)— v (i—=) 
Make the numerator equal toy, and the denominator equal to 2, 


which gives w= (7) = 1 --/z ; we have, therefore 
8 M y 


"dy «dz 
Scale 


y a 
E 


26 : THE DIFFERENTIAL CALOULUS. 
Mie * 
aim tate cotton 46 ms, — dx 
ey OE 2/0 a=) 


zdz 
vy Tayomay 
+dx : fd 


dx 
ia 


~ 2 (12?) 
yde Daal taie: 


a “ay (Imat) | 
whence we obtain, by ae eet stitution, ) bang : 
dy Redes: @ozdx 2 yd preg p22) da 
YL 7 yaw) 7 Ba eaw) — yea) * 

and observing, that 1? 422 = }./(1-+2) +f (ia)? 


+iv (+e) y=) tay 


fg TA Be gut TO Mi 


Taser Dx: > 
tive find 2 at last , 
du = 7 mm ; se aN ee 


The same result may be fou d in a. ee manner. : 
Let y = Ve and 2 = Aer)” ; theo, 


" dy+-dz — ne | 
du = ee 5 a 
ye yz ; ay 
| gy = 9. a ee Se 
va Yrm—mz x | 
Since y? —2? = BK —x) = 1+x2—1l+a= 27) 
ly 
But d ‘ and 2 
- rue) er a 
making these substitutions, the result i é » ‘ 
du = = Ita) 
Ex. 7. Leto = eye-n+yd 


r. en * peta Ty rua, ' re, . ring. Das ? ™ * i a >" “ 
. Lad Pile a > ¥ 

, tae a a | 

; . . + , as ae ~ : & ‘i ¥ » 

: . & 7 7 4 ea - 3 i - : 
~ 4. “ i x o “iy — 


wt! 
on THE DIFFERENTIAL caLcuLu Us. x 97 
hs 


Riv 


BI sic Sj. 089 7 = IEE VO): oh 
é taking the differential, we hae | Mee, Vat 


Fs ae ase ye i *. ‘ 
id 


-. Ais ® w= ee %, ge 


r. ¥ ie ‘ | i i 
z : ee hei dal ys 
<i * _? dnd } > 
aly : ‘ “ _ e . ee 
~ “Yue or i: . uf ee 
os) a mS +, me: 
“", € ? Tai dae a 
ee ee Viens 
= 


39, The po ® 


i) 


Pr ae of. = atials, 
Gon » Ex. ® Letty > 7 : 
Pe ‘Ting = 1 on 
ii al e oi. ast “ie 
. then “distdnhy we pnd F cao % in gpa 
* aoa “og N i 
| an hence du sad: z= : a te 
a a - i) i ; f, \ 


: q If in this ¢ case us nd 2 both = 
& ‘i eet t ni 
* y “ee 
Ex, 2.4 Let u = , 2, zand y being any funeti 


" | % | ie ‘ 
tabs Let y = 2%; tad oY, Bee: example, 
ce no tar . , 
5 poi & » ds == 4p Sy al rf, f 
iy eis oL J 4 oh 79 ar: : isha? 3 a 
con ae 
But dy’ = y WE + ledy " “Hence, iS oe 
7 i tae, 
* du = } at oy ly) <é | 


g ai : : ¢ % 
Dia ja x ae. 


| If in this case, v, z, and y ees then “=a, and 
’ " : J a. om . | i : 
Preah -. 


By means of these for : it will be easy. to find the ferential q 
of any exponential function whatever. ~ : 


i e 


a # 


és | hi Se 


'* 

4 PIRPERENTIAL CALCULUS. 
ciples in wis By ite shaogo: © ond a expel 

functio 8 Ae, 


= *, 38. ‘he. 


il. 


SP 3, Gl aunt gy arian tome a 


Bx 4 Let u= = ‘ / 


Ex. 


f ¢ oR 4 du= = ee 


te 10. hse a = (lx). | 
he Myf i i Ns a, 
f bya (xy 
a os ‘ : i RO 


5 Mel pety ss 4] ze 
ae ae “ie r x iL 
; ‘x, MM, Let = pao Na de 


* 
$ as Rie HRA 
“oz vs oo —z" , ig ae Me 
v ee \ wy? Qe ik ui ype 
YY M(t t 
wt ~ crn 


ae he, I v agar 
wi. ee Fic ae hte Ee 
Se du = = sp Tieiann * 


ee a 


ge = eV Meer). deeb ont 


ti Ge OE ht , 
: Me oer Pree or, 9 
hon ORME 


Messi eis Ry eh algo. 
aad Ex. 92. Let u = ev -D-he- VG ean A 


o~ oy ° j et . 
a i yi i ‘ fe e 
Fe net meer “ere: ah) wl 
. A +8 ~ a ‘ 
y 8 Sy. aK P ae 4 ® ; a 
: , RS YS” EE ic ota i 
- 2h ay 
yah. y a hah =. i 
we i 3 iy >, 4 ; rt 
ert 


air o WR india ey Le % a 


iy 
osines, hind other jrisoalienil Ee 


The Differe: iation of & Sines, C 
| i ay Gh, functions. 


‘d 
ire sevally enmed deehding the cha; 
refore the ares or angles : fy pres cos “ a 


ee 
\e 
al ’ 


34. Sines, cosines, 
acteristics, sin, cos, & 


t-. 


Cialis 
$i 


‘because 


30 ‘ THE DIFFERENTIAL CALCULUS. 


* x Se. denote the sine of az; cosine of x, &c,; the arc or 
angle being represented by x. 

Let « ibe" y represent any two arcs whatever, we have the follow- 
ing general property of their sine and cos, from thé) nature of ay 


circle. ipl , 
| om at ‘a 1 
Sin (ley) = sins cosy teostsiny. . . . - (a). 
cos (x+y) = cos % cos Pein: Sines 1 wy - 62) 


Let « = y; thensin 2a) sige cos vam: * x sin Z3 
_ SE chy. iM "3 i “ 


f ae 
in 2a = 2 sin b cos Le Wa 
Since “ may roppesent any are whatever, let 3 Ly ne substituted for 
vin the last formula, and we shall have “il | 
a “i Po. 
sin c= 2 sin 3 cos 17. ‘ 


Before we proceed to the differ Pr tion of circular ie it 
is proper to premise the following lemmas. 
it 


‘35. Lemna I. The limit of the ratios of aeons of a bifeilar 
are, and the arc ttself to the tangent, the are. being diminished with- 
out limit, ts a ratro of equadety ' ' 

For the sake of simplicity, we sha 
to unity ; then by trigonometry, 1 

. +. > 
chord  2sin ja 2 sin }2.cos ; 
Bae ¢ 


tans, » tape 


My ; 

cos xr i a A 
——— ; but sin a = 2 sin $a. cos iz. 
sinz © . 


: ee 


” be 7 ‘* 
! chord J FeOs & * j 


tang - cos$z 


ga taking the ligne when oes 05 


“. 42 = 05 and cos « =} cosx =.1 : whence the limit of the 
ratio of the chord to the atgent is a ratio of equality. i: 

Since the arc is included between the ch rd gand tangent, it is evi- 
dent that the limit of the ratio of it to either is a ratio of equality. 


~ Lemma Hl. The limit of the ratio of the sine of an arc to 


i Rosa arc re neeel both being infinitely diminished, is a ratio of equality. 


For if « be the are, we have Piatti. Ai 
. ORS 
* sin x 
cos 4 = ———. 
an 2 


* 
THE DIFFERENTIAL CALCULUS. _ di 


If c =0; thencos «= 1,3. .°. the limit of the ratio sin x to tan 
a, is a ratio of equality. ‘But since the limit of the arc to the tan- 
gent is a ratio of equality (35), it follows that the limit of the ra- 
tio of the sine to the arc is a ratio of equality. 
37. In order to find the differential coefficient of the sine of an 
x considered as a functior | of the are itself let “= sin x, and 


= singekd)s. erm 


et sin (xi 


by developing s sin (+4), and collecting the terms affected with sin 
x, We have : ; 


U 


But, by trig. cos i= 1 — 2 sin’ 37, and sin = = 2 g sin 1¢ cos ii ; 
, A | 


.. by substitution, + 


i— wv = sin mt 2s # —1)-12 sin 47 Xcos x cos 1i, 


~ 


P te a ‘ ee 4 M ee x 
or uw! — uw = 2 sin 12(cos x cos 47 — sin sin 10) : 
K yy. uk a ¢ ; s, 
* ul = w= 2sin Li X cos (+12), 
GE veccsendy Me abit Sg . 
aes Pia A 
Cd ome _ sin ae 
as 7 2H (cos ar 2). 
ee oe 1 ; 1a 


ey 
c hoe - 
tt 


i. 


i) 
oo, ae cos (x-+-31) = ¢ 


If i= 0, by (36), 


3 


die 


, ae ae =a cosa ; ‘ ae =="COS rdx. 


38. Having obtained this digeroatialy the dig@ion tia coeflicient 
of the cosine of anarce, considered as a function of the arc itself, 
may be easily deduced from it ; for we have the equation 


i ‘. wa 
° " & » 
sin’2* +c 


ro 


wy 


*¢ = lor (sin 2?+ (cos a) =]; 


“9 Me igh 
2 sin xd. sin x+2 cos «d. cos x = 0 ; . $b, No 
, oe @ ss Pee Ph 
whence we aehiva “ee : i al a eile 


* Sing or (sin «)2 signifies the square of sin x, and cos2 x pos u)2 signifies the 
wR 


square of cos. 2. sail ie, 


& 
~. 


a — we sin x (cos t— be ai sin 2 COs a. aa 


iffere ntiated, by the preceding ticle, gives..." je 


Se 


oT] 
32 THE DIFFERENTIAL CALCULUS. 


sin xd. sin x 
cos x 


d. cos 4 = — 


putting for d sin x its value cos rdx, and reducing, we have d. cos 
n=) gin cdz ; 


.. if u = cos 4, —=—= — sinz. 
dz 


This differential coefficient may be also found in the following 


; a 
manner: let vu = cos 2; then, since cos x = sin Sr a) 3* 
ant 
at sin(S — 2); .. by (38) 


T ae T 
_du = cos & — x) d. sin ‘3 —Z). 


u 


ML. So ‘ : ® F 
But 2 being constant, has no differential, and cos 5 —r) = sine ; 
therefore, ie 


’ pay du : 
du == — sin xdx, and ay nT Sinz. 
xz 


39. We obtain the differential coefficient of the tangent, by ob- 
serving that 


sin & 
u — tan. xr = f 


cos x” 
differentiating this equation, we shall find 


eet cos ad. sin « — sin ad. cos x 


cos’x 
But d. sin x = cos vdz, and d. cos x = — sin adz ; 
.. by substituting these values in the last equation, and observing 
the condition, sin’x-+cos’x = 1, the result is 


dx du 1 
cos’x ” dx cos’x 


: . a ee ° \ * 
And, since cot « = tan G — x) it follows from this that 


7 =2 semicircumfefence of a circle to radius unity. “ 


* 


THE DIFFERENTIAL CALCULUS. » 3 
dx dx 
d..cot= —*- meter re 
2( sin*x 
cos*?(—$ — 2 
2 
£ * 
1 4 
Or, if u= cotz =—~, wethave. 3 
tanz: 
t. ty ahs, 4 ty ¥ ’ 
 “d.tana dats dx 
du = d.cot 2 = — , “— — = - 
i) tana a tan? sin? 
ik. o> "er 


re (ts _° sim . ed y . 9 
for from the equation —— = tan, we derive cos tan = sin; .*, cos” 
cos 


x tan’x = sin*r. 


1 -_ eee 
40..'The equation sec « = ane being differentiated, gives 
OS x . 


d.cos% sinxdzx. snax 12 
d. sec ¢ => = aS ee a ee 


- cos2x cos2x coszcosx 


.. du = d.sec x = tanz sec xrdx. 


We can determine in like manner, the differential of the cosecant ; 
ae 


for cosec x = ——; differentiating we have 
. sin x ‘ye 
cos xdx ‘cosx 1 
‘d. cosec x = Rr PRE Ee ae ace - eg a Ly y cosec 


sin2z sin x sinx tan 
e ede 3° 


du = d. cosec c= — coi x cosec @ 


With respect to the versed sine of x, since vers x = 1—cos x, 


cS by differentiating we have 


d. vers « = see, x) = sin xdz. 


41. By the aid of these formule, we may find the differential of 
; any expression, involving sines, cosines, tangents, &c. For ex- 
ample ; let u = (cos a)so*, Make cos s =z, sin x = y; then 

> & = 24, and . ' 


oH ee 
i, oe 


du = d.z¥= 2 (dylz a :)) by (30); i 
fo a 
n ae ‘4 i 
. by substituting for dy and dz their values, — sin xdz and cos 
rdz, found in the preceding articles, we shall have. 7 
2 
; , se 2 
du = d.z¥ = (cosz)""*. ; cos x / cos x == sin'x is 
cos x | 


Fr 


dx. 


w 
pt s THE DIFFRRENTIAL CALCULUS. 


42. Having discussed the sine, cosine, &c. regarded as func- 
tions of the arc or angle ; we shall proceed to determine the diffe- 
rential of the arc considered as a function of its sine, cosine, &c. 
successively. 

The arc or angle, considered as a function of its sine, cosine, 

. &c. is usuaily denoted after this manner ; arc (sin = x) which is 
read thus: arc whose sine is x And arc (cos Cay denotes the 
arc whose cosine is x, &Xc. ee ‘5 
The arc or angle cons idered as a fnietion "Of at its sine, cosine, 
&c. is likewise denoted t us: sin—z signifies the are whose sine 
is x3 and cos—'a, the arc whose cosine is 2; Se. 
43. In order to differentiate an arc, conmaeral as a function of 
its sine or cosine, 
1°. Letu=sin-'2; .*, sin u = a, and, by (37), cos udu = 
dx, But cos u = 4/(1—2") ; 
. i he dx 
ne uu Vie) 


‘ 2°. Let u = cos—'x. Since, sin—'z-+-cos—!z = ga 


. a. sin—'z-+d. cos—'z = 0, 


* 


d. cos—iz == — d. sin—'z. 
dx 
v(1=a7) 

Therefore, since x being the sine of an arc, ,/(1—<)? is its co- 
sine, and x being the cosine, ,/(1—<”) is its sine; there results 
from what we have just found, that the differential coefficient of an 
arc expressed by its sine, 1s equal to unity divided by the cosine ; 
and that the differential coefficient of an arc expressed by its cosine, 
is equal to unity divided by the sine, and taken with the sine minus. 

44, To differentiate an are as a function of its. tangent or co- 


ff dus d. cost = — 


tangent : “? 
1% Letwstan's: |. tanu=—27; . © 
. 3 du “a? 
pea “. by ( 9), ae dx. » 
2 a ~ Butsi since 
‘a gO READS eee 
‘ ~~ sect P cee ia, 


A 
. 
" ; 7 A) 
THE DIFFERENTIAL CALCULUS. rss 
‘< T 
2. Letu=cot—'z. Since, tan-'s-+cot—'s = a 
> x” 
Hes d. cot—!z ie d, tan—'z, 
. » te! 
and consequently ; 
aps 
dx 1+2? 


. - 

Hence, it follows, that the differential cficient of an arc express- 
ed by its tangent, is equal to unity divided by the square of its se- 
cant ; and that the differential coefficient of an arc as a function of 
wts cotangent, is equal to unity divided « the square of its secant, 


and taken with es negative sine. inte 

45. In order to differentiate the arc as a function of its secant or 
cosecant : & 

1° Let w= sec!z ; then, sec u = 2; as 


~. tan u. see u. dus dx by (40) 
But, since sec u = x, 
tan a = /(x°—1) ; 
du = ee or be os ? Se 
a de EJ (G1) dz t/t" —T) 


is 


og , a oY 
2°. Let u == ¢osec—'z. Since sec—'x-++cosec—'x = ‘ 


an 


“. d. cosec'x = — d. sec—!x 
oe 
é * ; dx he 7a 1 
cK i. = = - —; : 
ayia Nae af (x21) 
46. With veneet to the arc considered as a fanction of its vers- 
ed sine. Soe i 


P Let u = vers—'x ; then vers u =z ; 
— sinudu = dx. 
But sin u =,/(2z— 2?) ; 
dx du yd; 
VQinay i V@aaz) 


47. We shall conclude this section with examples of the appli- 
cation of the rules for the differentiation of circular functions. 


Ps du — 


Ex. 1. Let u= cos nr’; 


36 \) THE DIFFERENTIAL CALCULUS. 


“, du = — sinmz-d(mx), by (38) ; Hence, 


OT ig ‘ 
fe m sinmx. | ; 
Ex. 2. Let wu = sin (qm) 
, “5 du = cos fom at (a) ), by (37) ; 


but d(am) = a Hence, 
ahs du 
m—1 ™ 
ene poate )e 
_ Ex. 3. Let u = cos ASM ~1) a Lie 


du : 
“. 57 = -— sin z+,/(—1) cos 2, 
dx a 


or = Vv (—1)-$cos a+,/(—1)sin cf. 


Hence in this case ace = 4/(-—1).udz. It appears from this and 
, iy example 22* in the last section, that the differentials of the function 
ev (—-), and the above are the same. It will also appear from the 
- integral catealns that these functions ; are actually equal. 


Ex. 4. hel » j= as 


. du = cos xd. sin x+ sin xd. cos 5 


which, by sclisbibiee for d. sin x and d. cos x, their peluct, gives 


4 


= = cos’s — sin*z = cos 22, by (34) ; 


for if we make x = y, in “formula fa (2), we shall have cos 2x = 
cos*x—sin’z. 


er 
ee 


Ex. 5. Let « = sin x cos a+ sin a cos 2; 


, du = cosad. sin x sina d. cose, 


oe = cosa cos x= sina sin x 


The differential and integral calculus is of very extensive use in 
the deduction of the formule one from another in Trigonometry. 
For example, in the last, 
| u = sin(rzta), »., du = cos (xta)dz, and 
« oa 

du 

— == cos (x#a).. 

rs (Ea) 


eg a aa er 
* There is an error in the answer to the. example just alluded to, it ought tobe, 
das es feOV Ee OM hy (1). 


x 


THE DIFFERENTIAL CALCULUS. 


Hence cos (xa) = cos x cos a = sin x sina. 


Ex. 6. Let vu = sin 22, *, du = 
' d 


ater 2 e I= 
—_> = cos 2x. 
dz ma i th | ‘ 


‘cos 2xdx, and 


By this and Ex. 4, it Nong that if sin 2x = 2 sin x Cos z, 


cos 27 = cos*x — sin2sr. 


Pee 


37 


Ex. 7.. Let be an arc, or an angle, whose sine is the function 


e ' ie 
v a Po make ne = z, and we shall have dp = 
dz er 
& : pe 2] 
Fn if a om 
ee SiN I al 
a | 
aK ty ae 43 Xe, 
ar ae ge Jf 27)’ 
1- Qy2 
and 4/(1—z a 7): 


ai 


% a - T=V0-) 


A oneal goat y) 
make 2u (l—u?) =z; then, dr = 


de 
SJ 2) 
Ey Sand — 20 and /(1—z?) = 1 — ou? : 


“Jae 


a 
V7 (1—u?) 
Ex. +9: hive 2 aos x sin 22. ; 
du = (cos x cos 22 + cos 3x)dzx. 
Ex. 10. Let uv = (tan x)”. 
du _ n(tan x)"~" 
dz (cos «)* 
Ex. 11. Let u = I. sin x. : 
: du = cot xdx. 


‘ . oe aS ws 
* ie “den a “i 


e | 
: , P v 
nt ay Wy MMR SE Sak det 2 gta 2 
A cul : . : . or A= 
t mee ." AY / . 
Ye 
‘ ~ 
? 7 *~,. 
: 


Ex. 14. Let u = etme, dus eins cos xd. 
ths \ & hs Pa je oo Ky ae 


Ex. Ws: Let c= vie’ ik ee sin; 


jae Urey uaa bs if i 
due fas ‘ - 
WEY diel: 4 4 4 . : 
‘ ar = ieee 4 . Sl : : * tx 

hall PA gy btitg x ih, ha en a. te J rund ( t 
ie. ees Letu = = cos (In pe? ae — sin. ae 

* we ; ; ahs teks: Me Nt ad + a | 
*; BH 17. Let ws sin-i( 1 +3 ve A or te ot at vt 
% “ > ae 


J si a ‘ig | 
, u = are cn ae — nt) 


is v i Mi tidy ey a “he a Sih nae ila a ms, } 
iis “iG Se Qdz ‘:, Ma et * 
> o ; ‘ r ¢ Ket du= =e ade ‘ ‘ é 
; sn i . a fy a, bid sl : 3 " . 
(et a a re ) . 
th ‘img i ' rie iy wae TR i. * OM, ae ® 
Ex. 18, Late! = sin v( 3 9g du 
g ie x. pe es a of ei) hin 
Ex. 19. Let u = = crue Ba"). at F r. | 


ahh 2 OS =8de_ 
he O/C Sat: 


20. Let w = sin! 


pes aE 
v (1+2)" 


2 


— 


hee 
' : ee! ne a 


Pi (xn ke 
22. Let x i geal se 
\2na— 1)" 


| THE DIFFERENTIAL CALCULUS. 39 
v yy 


oh 2) 
Ex, 23, Letu = tan—! eee! 3 


z 


ee. Let u = tan~'$// CT") i es 4 2Ga a? 


* 
SECTION V. 


-—_ Successive Differentiation. 

48. In the several functions. which have been differentiated, it 
may be observed, that the differential coefficient is a function of 
the variable in general different from the primitive function. This 
function therefore itself may be differentiated, and another diffe- , 
rential coefficient will be thus determined, which is called the se- — 
sond differential coefficient of the primitive function. As me 
GRANGs calls the first differential coefficient the first derived func- — 
tton, so he calls the second differential coefficient the second deriv- 
ed function. From what has been said, it is plain that the second 
differential coefficient of the primitive function is the differential co- 
efficient of the first differential coefficient considered as a function 
of the original variable. 

The second differential coefficient, like the first, being in gene- 
ral a function of the original variable, is susceptible of differentia- 
tion, from whence results a third differential coefficient, or, accord- 
ing to Lacrance, a third derived function. By continuing these 
differentiations one after another, we deduce from the proposed 
function a series of limits or of differential coefficients, which are 
distinguished into orders, according to the number of differentia- 
tions which have taken place in order to obtainthem. Let u = fz, 
and if we make — e: Bie 


du dp dq 
ae p, Be qo = 7; Ke. 


then p will represent the first differential coefficient of the proposed 
function, q the second differential coefficient, r the third differen- 
tial coefficient, &c. Now, since g is obtained by two successive 
differentiations, and pel each time by dx, let us represent that 


tg 
~ 


40 ‘ THE DIFFERENTIAL CALCULUS. 


2, 


i a «. Guy 
operation by _ and we shall have e 


=q; in like manner, by 


again and dividing by dx, we vm all have =r, &c. 
“so that, ote | . ik 
am _ dwis the first differential ard ihe 
~ du is the second differential ; 
@u is the third differential ; . 
&c. ’ 

It should be remembered, that the exponent which accoripanies 
the characteristic d, indicates the repetition of an operation, and 
not a power of the letter d, which is never ponies as a quanti- 
ty, but merely asa sign. ‘iin tee * 

The expression dx’ signifies the square of or (dx)?, and not’ 
the differential of x°, which is usually Mine dia? or d(a?) ; 
again, dz signifies the cube of dx or (dx)°, and soon. ; 

49. If, for example, the proposed function was az”, we sbplald 
find d.ax” = nax"—'dx, by (23) ; the factors na and dx being con- 
- sidered as constant in the first differential naxz"—dzx (since it is the 
- differential coefficient nax"—! which we. are to differentiate,) it is 
sufficient, in order. to obtain the second differential, to differentiate 
a*—1 and to multiply the result by nade ; 5 but d.2* 7! = (w—1)a"-? 
dz; we have therefore, : 


* 
eee 
ay ‘Bh ; 


d? ax” = n(n— Iaa*—2da%. 
tn a similar manner we should find , ty oe 
B.axc* = n(n— 1\(n—2)aa"—Sdz3 P 


d‘.ax" = n(n —1)(n—2)(n —3)aa"—4 da, 
&c. Xe. 


a  s 
ay 


and if u = az”, the diferente coefficients woul ave the follow- 
ing values : . ns 


du | 

a = nax—! 

a = nn—V) ax"--2% 

= ie eae = pes i 
pt ae =1\(n—~ way 3)aan—3 ! és 


i be! xo. 


THE DIFFERENTIAL CALCULUS. ¥ 4j 


Jt is obvious that when the exponent is a whole positive num- 
ber, the function ax" has only a oe number of differentials, the 
highest of which is 
| ma dan" = n(n—1)(n—2). 2... 2.1. adz™. . 
This expression is no longer capable of differentiation, since it no 
longer involves a variable quantity : we have then for the last diffe- 
rential coefficient 


a” azn 


Gia n(n—I1)(n—2). 2. + 2 Lada, 


which is a constant quantity. - 

50. This remark furnishes a very simple method of developing 
in a series according to the whole positive powers of x, any func- 
tion u of that variable, provided the development can be effected. 
If we assume the equation 


a= A+ Barter’ +-De+Ez'+, &e. 3 
by differentiation, we find if 


te 


Bh = B+2Cr+-Bde?-+4Ew-, &e. 
_ 

ea SE 2C+2 3De-+3.4E x? 4, &e. 
d?u 

“og = 1. 9.3D-+-2.3.4 Ex-+, &c. 


&e. &ce. 


Let (w) represent what « becomes, when 2 = 0; 


du du 

bes = ee 
a2) DS de af ; 
a? 4 > . d2 4 8 Bay 
Daadece  amaauaeet | 


and so on, the preceding equations give 


() = 4 (Ge) = Bes) $26 (Gs) = 280, Be 


from whence we derive | 
G 


A2 ‘ THE DIFFERENTIAL CALCULUS, 
a= (v),B=(o cee sf presets &e 
, enn; ~ Q igs) 8.86 oa) sa 


substituting these values in the primitive equation, it will become 


d2u 


w= ()+(Z) 3 (as)* wt os : 5(Z:) not, &e. 


This formula is called Maclaurin’s Theorem, because it was 
first given by that celebrated mathematician in his Fluaions. 

51. For its application : if we take u = (a+<a)", we shall have, 
by differentiation, 


du - 
aa =< node f My ie 
dy 
oe ain noDlatsyt 

” # 
[3 
ot = n(n—1)(n —2)(a--a)*-5, 


&e.. &c. 


and making « = 0, the value of wis reduced to a» ; therefore, 
(w) == a”, and the differential pesrniger become, 


=) = ee xe * - ee Kc. 


Substituting these values of (uv), (= a 4) & y, ae in Maclaurin’s 
Theorem, (50), we find 


n—1 n—2 


od 


“= (a-a)* = a"t-na—le-Len,— Thee 


a—3y3 +; a Ce 


soe 


The principles of differentiation having’ been | deduced without 
assuming the development of (a-++-)", we may consider this as 
proved for all cases, whether the exponent be integral or fraction- 
al, positive or negative: In common Algebra, the expansion of 
(a--2) is called the binomial theorem. ke 

It is evident that Maclaurin’s Theorem will furnish the means of 
developing any funetion of « whatever, in terms of ascending inte- 
ger powers of 2, and constant coefficients, if the function itself be 
capable of such a form; but if fx be of such a nature, as to in- 
volve in its development, negative or fractional powers of x, it will 


THE DIFFERENTIAL CALCULUS. 43 


then be found, that this theorem will fail in effecting it : thus, if the 
; 1 1 

function to be expanded be am the first term (w), being g ois infi- 
nite, and the function cannot be expanded in the required form. 
This, however, ought not to be called a fault or failure in the theo- 
rem, because in these cases the function does not admit of an ex- 
pansion in the positive integral powers of the variable. 

In like manner, if the function was wu = cot 2, the first term of 
Maclaurin’s Series would become infinite, since, it would be equal 


1 du 1 du 
to——; for — = — ———~ (39), and whenz =0 (;-) ge 
0’ dx sin? a ( ds P > Ada 
I 4 
- = = ©, « : ral é> 
0 ® " fi Ys 


ei 


52. If in any funetion wu of x, the variable x is changed into 
«-+i, the differential coefficients fermed by considering x as varia- 
ble and 7 constant, are the same which would be found by treating 
2 as variable, and x as constant. In order to demonstrate this, if 
in the — u = fx, we put x-++7 = 2’ in the place of x, we 
shall have u’ = fz' ; the differential of fx’ shall be another function 
of x’ represented by $2’ and multiplied by dx’, consequently 


du’ = oa'dz’, 
or by putting for x’ ” its value x-+i, we shall have 
| ad 29241) d.(a-ti). 
Now, on the supposition of x alone to vary, we shall have 
dl = (a-+i)de, 


whence we derive 


du’ 
eet) seg aati ht. nays 8 


If, on the contrary, we make x constant and 7 variable, the factor 
d.(#~++7) is reduced to di, and we obtain 
fe 


Tabs aes Pet) eek Ee 
hence by making the two aialacs of o(x-+2), equal to each other, 
we have 


dues a 
dx di 


Seb THE DIFFERENTIAL CALCULUS. 


For example, if we had, uv = ac’, by putting 2-7 for «, we should 
find 


| oor iki FA “= Ba(e-+i)? ; 


and consequently 


dx ae 


The equations (1) a (2) borne differentiated, give still the 


equal results 
it, 
CL 6 ae Rx 
“de =©0 Canela 


Oe = g(e-bidle+i). 


Making 7 constant in the first equation, a and x constant in the se- 
cond, we shall have 


z i 
d2 uy’ du 
Tag = ‘(ae ) dx, “de = bia 2 
from whence we derive 
Pu du a 
a a! 


We shall conclude by the same reasoning that 


d3u d3uo d1u dtu 


a= 


NE oe So on. 
dx? dis’ dz de? 


53. This being premised, let uv’ = Ha-ti) 3 this bells supposed 
to be developed according to the ascending positive and integral 
powers of 7, which can be always effected (9), provided we do not. 


assign any particular value to x, and we shall have an equation of 
the form 


a = utAi+Bi2+CP+, &e. . . . - (1). 


A, B, C, &c. being unknown but determinate functions of x. In 
order to determine the values of these coefficients, we shall in the 


first place differentiate this equation with regard to 2, and dividing 
by dt, and we shall have 


THE DIFFERENTIAL CALCULUS, 45 


, 


“ = Af 2Bi4+3CP-+, &e. ; 


differentiating aferyasc with respect to x, and ere by dx, we 
shall have 


> 


“din dasegde Pott, Be 


The first members of these two equations being equal (52), the 
second members shall be identical ; equating therefore each term 
of these two results, we shall find 


du a eB eae 
A= a ag isan? = ap &°- 
Substituting the value of .1, given by the first of these sa ati, 
in the second, we shall have 
Satara 
“412 det’ 
substituting this value in that of C, we shall have 
fe 
to aes 
1.2.3 dx3”’ 
By means of these values of 4, °B, C, &c. equation (1) will be- 
come 


ome 


d?74%4 ,d°u 23 
w = f(a-+i) aut S “i+ ta ae isa By —--, &c. 

This formula is called Taylor’s Theorem, from the name of 
Brook Taytor, by whom it was first discovered : he does not ap- 
pear to have been aware of its importance, but dismisses it without 
remark in a Corollary to the Proposition from which it is deduced, 
which was first given in his Method of Increments, published in 
1715. a 

It contains implicitly the development of the binomial, for if we 
suppose u = 2”, u’ will become aba and we shall have 


(a--i)" = ab na®—li-bn(n—1 a" 


+, Ro! 


54, The formula of Taylor shows, that the various differential 
coefficients possess the remarkable property of forming when they 
are respectively divided by the products 


5 tn(n— 1)(n— 2) aS 


46 THE DIFFERENTIAL CALCULUS, 


1.1.2). 1.2.8) &e: 


the coefficients of the powers of the increment t, in the complete 
development of the difference 


duit .d?u i . dy 73 


pat = oe tg.s ia he TS 


dx 1 dx? 1.2 Last? &- 


This development, when we make Au = uv —u, andi = Az, and 
the arbitrary quantity dx be supposed to equal Ax, becomes 


du d2u . d2u d4u 


Au= Tie Tesh peo4 


1 12, 128 “hy &e. 


which expresses the difference of the function in a series of its cor- 
responding successive differentials. ‘The chataeter A before a 
variable, signifies its finite difference. 
55. As an application of Taylor’s Theorem, let us determine the 
the development of uw = a*: we shall have, by differentiation, (29), 
du ; d2u f d3u 
ea ee oa © ake oo age eat, Sc. 
Therefore, by substituting these values in the development of 
f(a+-7), (53), we shall have 


act! ee a*-beai+s ae 


ports 8 &e., 
and dividing by a”, 
= 1404+ oe +e. Ak or a CRE 


The first two terms of this series has been found (29) ; by 
means of it we may develope any exponential quantity, arranged ac- 
cording to the powers of its exponent : for, since 7 is an arbitrary 
quantity, we can substitute x in its place, and then we shall have 


cx? e343 
ee 1-pea-+—> Teg bite. soe ee (2)5 


in which x may be any quantity whatever. 

We may, by series (1) or (2), determine directly the value of ain 
terms of c. 

By making i = 0, we shall have 


m a= Ite+— seas ty Ke. 


THE DIFFERENTIAL CALCULUS. 47 


And when c=1, the value of @ is found to be expressed by the 
very simple series 


244 Stag teagt &e 


whose value to seven places of decimals, is 
. 2.71828 18. 


This is the number which is usually designated by e,(30), and 
which is consequently the base of the exponential whose modulus 
is unity. 


: ee | 
If in the equation (1), we make z =~, we shall have 


1 


= 


ae =1+1+5 ites) — aCy = 


Thus we have among the quantities a, c, and e, the relation 


as 
aes; wae 


This equation gives also 


hence we see that by taking ¢ negative, a becomes ~ —; thus, by 
making these changes in the equation (29), 


(a—1) , (a—! Ny 


e=a—-1— 5 3 


— &c. 


we shail have 


€ mot (a—1)° 


3g te & 


a—1)? 
a Gos. 
This series is more convenient than the preceding for finding the 


value of c when ais a number greater than unity. By making « 
= 10, the base of the common mem of logarithms, 


2 ey ey. same ae 


hence, we shall find by calculation, 


¢ = 2302585093, nearly ; % 


48 THE DIF FERENTIAL CALCULUS. 


1x. 
2.302585093 


= 0.434294482. 


] 
"4 = 


This is the value of m, the constant multiplier ‘of the differential 
coefficient of log u, (29): it is also the modulus of the common 
system of logarithms, or, which amounts to the same thing, it is the 
number by which we multiply a Naperian logarithm in order to pass 
to the corresponding common logarithm. 

For, since ae" = e’*, (29), and a = e°, we have 


ee log « prea eles Ke. c log a la ~ 


and consequently, 


I 
log x = ial = mlx. 


SECTION VI. 


Application of ‘the formule of Maclaurin and Taylor to the de- 
velopment of Algebraical and Transcendental functions. 


56. It may be observed here that the series of Maclaurin may 
be derived from the more general series of Taylor. For, since 
5 i om oe dee} eles FB eae. 2° 
d flet) —e tare Waa 


Let (uw) represent what u becomes, when x = 0, 


ip 
du du Oo” 
=) s e > ° ° e hi e x —— 0, 
a2 aL d? Uu a 
=) e . ‘ e CY dx? e & — 0, 


and so on, for the other differential coefficients ; the formula. of 
Taylor, when we make x = 0, will become therefore 


du,., d?u.v? | ,d3 , 
fi = (W)+(Z it Gast Gea)-agt &e- 


When x == 0, u = f7, and therefore the differential coefficients of 


* 


THE DIFFERENTIAL CALCULUS. 49 


this function must be the same functions of 7 as those of fx are of 
«. Hence it follows that the latter, whenz = 0, become identical 
with the former when 7 = 0, changing 7 into ey xe get 


d2u de dau 
fe =u = (w+ (Z)e+ ta) (23 
which is the formula of Maclaurin.* i 


57. Let uv = J (a+?) ; we have therefore u = ,/z = x? ; 
.. by differentiation, 


BY 2s 


= 


du we 1 
— =F —Z = ; 
dz. 2 Qf x 
dtu im 1-2 he. i 
® TS ey Stet” Wy eg 
iqene’ 3? § 3 
reg aE rr ia 


substituting in the formula of Taylor, we have 
UD e 
yu = J (x&-+t) me vitg Sya?! i625" &e 


58. Let vu’ = sin (x+2) ; hence it follows that u = sin x : we 
may therefore find the successive differential coefficients of u = 
sin x; thus, since, (37), du = cos xdz, we shall have 


du d2u, dea d4y & 
cosz, ——- = —sinz 5 = = cosz = = sin x, C. 
dx , dx? ; da? > dx4 


substituting these values in 'Taylor’s formula, we find 


* ; m a : ar Oh 
n -7) = sin x-+ cos x.— — sin 1.——— — GOS x. wf 
Pe Ff he °5 - 
sin x. cos &c 
naan hy i 


making « = 0, then sin « = 0 and cos x = 1, and this develop- 
ment becomes 


. a3 ? 
© i } = } aan Se Tee 2 ° ° e e ry >, 
leas aM ae. 


‘ . => Hey FP elec e 
* Lacroiz, in his Elementary Treatise on the Differential and Internal caleulas, de. 


rives the formula of Taylor from that of Maclaurin. 


50. THE DIFFERENTIAL CALCULUS. 


If we take uw = cos(z-+7) ; and therefore wu == cosa; by pro- 
ceeding as in the last example, we shall find 


a2 
eo iy ee 
cos 7 = r. wer: 


ha qo : 
me a ae caer Ty ot 
2.3.4 2.3.4.5.6. 
The two series Ay and (2) may be readily deduced from the de- 


velopment of sin (z-ht)s for arranging this by the factors sin x, cos 
x, we obtain 


: 28 Mts 4 
. v ‘\ ——, = 4 pe — ora. oo e 5 
aa sal ag Stage 3.4, ~ 2.3.4.5.6 e 
z e ay ' ¢ 
COS x 1 a 232345 ols ee + Sige e 
But by trigonometry, -- 


sin («-+2) = sin x cos 2 -+ sin7 cos x. 


Since the value of 2 is independent of z, the equations must hold 
for all values of « ; hence, by comparing the corresponding terms 
of the last two equations we shall find the two series just al- 


luded to. These series might be also deduced by Maclaurin’s 
theorem. 


du. 
59. Letuw = tanz; .. — = sec’; hence | 
dz 
d2u ! As 
—- = 2 sec ad.sec x = 2 sec’ tanrdz. 
dx? . : 
d?u 5 : 
a 2 tan 2(1+tan’x) = 2u(1+u7). 


Hence the third differential is 


d3u 
— = 2 
a = 2(1+3u )du ; 
but du = (1-Fu")dz ; ; 
du 
4 _=—9 2 2) 
Sa Pe ate (1--u?)(1+3u") 


By continuing this process, the succeeding coefficients may in 
like manner be found. Substituting in Maclaurin’s series (50) 
for the coefficients the values of the differential coefficients of 
this function, the result is 


THE DIFFERENTIAL CALCULUS. 51 


2x5 2*x% 
actions auth XC: v" 


x 
t ——— 
ae i*T 2.3 1.2.5.4.5 


The differential coefficients of the cot « may be deduced from 
those of the tangent by changing x into é —2), ana changing the 


the sign of dz ; and consequently, the development cot x can be 
readily obtained. 

59. Let u = sin—!z ; it is required to express the arc u in a se- 
ries of powers of its sine: 

By differentiating successively, the first differential coefficient 


d 
<= (Len?) 4, (48), we find 
d?4 i 
“> ania Tay 
axe 
du = of | nae 
Pag, Ge ma) 7. 24-32°(1—2*) 2 5 


and so the process might be continued. 
Substituting for the coefficients in Maclaurin’s Series what these 
differential coefficients become when x = 0, the result is 


sin?u . 32sindu 


u = sin ut-——_ TX y CL &ce. 

If wu = cos—'z, the successive differential coefficients have the 
same form but different signs; consequently the development of 
the arc win terms of its cosine may be readily obtained. 

60. Let u = tan—'!x; to express the circular arc uw ina series 
of powers of its tangents. 

Let the first differential coefficient 


du an 
ag C2") 
be differentiated successively, we shall find ea 
d?u 
ike — 2 t+ 22)- 24-2.4.27(1-fa?)> 3 
i;? A. 
dtu O3 2 3 pi é 3 2\— 4 
ae 9733 X(1-Pa?)— 9 — 24.3.2°(1-b2*)- &. 


: 


52 THE DIFFERENTIAL CALCULUS, 


And in this manner the process may be continued. Substituting 
‘in Maclaurin’s theorem the values which the ‘differential coefli- 
cients, thus found, assume when x = 0, the result is 


faa tan u tan*w _ tan>u tan'u 
Yaeer y ON SER 7 
® a 
Cor. If u ee oman aa 1S gn; 
eS 1 dl ] ef 
= 4 me te le bie 
Se ee 


This series is not sufficiently convergent for the purpose of com- 
puting the circumference of a circle. - 


61. To expand the function /(~-++2) in a series of powers of 7. 


Let « = Ix and wv =l(x+7) ; .*. by differentiation (31), 


- ie Hence, by successive differentiations 
x < 
ad? u APS vee 
Fer TS ee RRR ah 
dy, 2 
dx? = ar = =H aps 
d4 2.3 
oy: = areas © ESOT wets 9.3 yn} . 
dx* ge 
Sc, we. 
substituting these values in the formula of Taylor, we shall have 
: ; ; ae.) dee a 3 
att) = cl eee 
72 23 74 


z 
ae is ry —l. =>=-—> —-— —— — ——, 
Te x O52} 3q3 4x! 

When 7 is small compared with 2, this series converges rapidly, 
and therefore serves, when the logarithm of one number is known, 
to compute the logarithms of a series which varies by a very 
small difference. 

If in this series « = 1, it becomes 

Bi geet: ae 
l DN ee ies se emg 
1) Se a Bor od 


which, when 7 is negative, becomes 


THE DIFFERENTIAL CALCULUS, 58 


: z e ae q? 
ema ance 
Hence by Subtraction 
1+2 ae 
(==) =2 ght mr .gotiet ¢ 
amis I+; ni oe 


Hence the last series becomes 


(n-b2) Re ) +, ke. 


Q Senses | Cane po Biss 

jer pre) arate 
this gives the logarithm of n-+-z when that of x is known: let 
=1 and z =1; hence 


l H 


3.32 3.32 53s TiT3gT 


2 = aye _ rts Se. § 


Tot 


this series rapidly converges, and therefore gives the hyperbolic 
logarithm of 2 to any required degree of accuracy ; its value to’ 
seven places of decimals, is 12 = 0.6931472. For larger num- 
bers it is still more convergent. 


SECTION VII. 
Differentiation of Equations of two variables. 


61. When an equation 
Pm OO eG). 


involving two variables, is given, either variable may be consider- 
ed as an implicit function of the other. By resolving this equa- 
tion with respect to y, we shall find an equation of the form 


” y = Oe. 
In this state the function might be differentiated by the rules 


54 THE DIFFERENTIAL CALCULUS. 


already given for the differentiation of functions of one variable, 
and thence the value of the successive differential coefficients 


dy d?y 
AG? ie bee, &c. found. 


For instance, lets 


f(a, y) = e+3ay—y =0, . . . (2). 
by resolving this equation, we find 


y= Sv Get) we A pe 0: 


-, by differentiation, and taking the positive value, we have 


fl = . (4) 

za? pa) 

Differentiating again and dividing by dz, we shall find 
kN fy (5). 


TAC ee aN ee 


By continuing the differentiation, we shall find the successive 
differential coefficients of the proposed function. 

This method, however, would in general be of no practical use, 
as it would require the general resolution of equations which can- 
not in general be effected except by methods of approximation. 

62. It will be therefore necessary to find the successive differen- 
tial coefficients of y with respect to x, without effecting the resolu- 
tion of the proposed equation. 

For this purpose, if we conceive the saiaon resolved with re- 
spect to y, we shall have, y = gz, substituting this value of y in the 
proposed equation, f(z, y) = 0, it will become 


S(@ ox) = 0, 
an equation consisting of the single variable x, which may be, 
therefore, represented by 


This equation must evidently be identical, that is to say, it must be 
composed of terms which destroy each other, whatever may be the 
value of x, and consequently also by putting «+7 instead of <x, 
whatever may be the value of 2. | 

By substituting «+7 for x in this equation, we shall have by Tay- 
lor’s theorem, 


THE DIFFERENTIAL CALCULUS. 53 


a.Fr., &uFx 7? DE S 


F(a-+1)=0=Fe+— BMAP Pm 2 a ae &ce. 
and tet ie since the increment 7 may have any value what- 
ever, . 
d. Fx Ca ew dix 
Fz = 0, Pe ae a 10s 5 =e On tee. 


or, replacing Fx by f(x, y), we have 
hey) = 0, ED) a FO 
f(t, y) = 9, a 0 ae &e. 


b) 


Hence, if we regard y as a function of x, by differentiating the 
equation f(x, y), and putting the result equal to zero, we shall be 


able to find the value of the differential coefficient iy 


Let take, for instance, the equation, (61), e 
f(y) = o-+3ay— —y =0; 
from which, by differentiation, we obtain 
ft Qxnde+-Bady — 2ydy=0; . . . (7); 

from this equation we derive 

Ee ea 

dx 2y—3sa 
The value of this differential coefficient is given in terms of « and 
y ; if it were necessary to obtain 4 in terms of x alone, we must 


substitute in this expression the value of y (3), deduced from the 
proposed equation ; this substitution, by taking the sign plus, gives 


dy Qu Ww x "a 
dz WE RO eo Mie. i 


= 9 
pe _ 2 2 tbe 72 2 
a(3 rie ete) -380 (5"°+2’) 
the same result that has been already obtained by differentiating the 
equation (3). 
63. In order to obtain the differential coefficient of the second 


d? ; in 
order, that is to say, the value of 2, let equation (7) be divided by 
GL 
| o dy ; . ° 
dx, and making == p, that equation will become 
ae 


22-+-3ap—2yp = 05° se. “ae (&) - 


56 THE DIFFERENTIAL CALCULUS. 


differentiating this equation, considering y. and p as functions of 
x, we shall have © 


2dz+3adp —2ydp—2pdy = 0 ; 
dividing by dz, and putting p instead of a it will become 
2-}- Base ~ ay — 2p* = 0, 


whence we derive 


dp = 2p? ~ —2 . 
dt. pias = By e ny ° e H s e ° j (9). 
dy d d? 
Now, since p = = we shall have oe = 3 : putting these values 


in the above equation, and taking away the pee we 
shall obtain 


“ @y(3a—2y) = 2dy? —Qde2 sew (18), 
an equation which expresses the relation which the differential co- 


dad? 4 
efficient of the second order, — has to that of the first order 


of and the variables x and y. 
ax 


; d?y. 
We might deduce from this equation the value of 4 in terms of 
ei dy? ; Nace 
x alone, by substituting for Tap? and y, their values found in equa- 
tions (3) and (4), and the result would give equation (5). In or- 
Aid Se d 
der to find the third differential coeflicient, make ‘- = g, andequa- 
tion (9), by clearing of fractions, will become 
3aq—2yq = 2p? —2; 
differentiating this equation, and regarding y, p, g, as functions of 
x, we shall find the third differential coefficient, and so on. 
Instead of employing the letters p, q, 7, &c. for effecting the 
operations, we would arrive at the same result by differentiating 


equation (7), and putting dy for the differential of y, d?y for that of 


y, d°y for that of ay, &c., and considering dz as constant ; we 
should find 


THE DIFFERENTIAL CALCULUS. 57 
Lda? 4-3ad°y — Wy? — 2yd?y = 


which is the same as equation (10). 
Hence, it appears, that the successive differentials of any proposed 
primitive equation may be deduced one from another by successive 
differentiation, regarding y, dy, d’y, &c. as functions of x. 
64. The equation (2) being of the second degree, affords two 
values of y, in consequence of which the equation (7), gives for 


d : 
the differential coefficient —2, two values corresponding to those of 


dx. oi 5th) 
the function y. : | 
If, instead of resolving the proposed equation to determine the 


values of y, we eliminate that variable between the two equations 
a? Say — ¥? = 
Qedu-+-3ady—2ydy = 0, 
from the second of these we find 


_. 2adz+-3ady n 
ae) 2 ae 


and substituting this in the first, ad making the proper reductions, 
we find 


dy? 3 
pag & Tones} 
dx? Sa*--x 


te Pa enced — sa ia ile 
de Qa 


65. It is easy to apply the preceding method to much more com- 
plicated examples, or to Seine in which the variables rise to a high- 
er degree. 

Take, for instance, the equation 


f@. 9) = 7 Baayte=0; °°. . (1); 
from which, by differentiation, we obtain 
3y°"dy—Saxdy—Saydx+-3a'dx = 0, . . . (2), 


and consequently 


The function 1, in this'example, being givenby means of an equation 
I 


58 THE DIFFERENTIAL GhieLos. 
%: 


of the third degree, ought to have three values ; and by substitut- 


dy» 
ing them sucessively in the expression for a we ee obtain an 
fe 
ba ven a 


equal aber ‘of elves, a this differential koeeeiene We may 
observe, in general, that this coefficient will have as many different 
values as the function y has in the original equation ; ; and the same 
holds goed with regard to the differential itself. i , 

If we eliminate y between the two equations (1) and: (2), the re- 
sult would be an equation of the third degree, with respect to dy ; ; 
and it would contain, as Als gor, the three steht of tae et dif- 


ferential is susceptible. be at 


re gies 


Having found the expression for: dy, or that of ae we “ay obtain 


i x 


ae 


ey 4 ii 


" s by differentiating the first differential (2) of 


those of d? yor = 


the proposed onal with respect to dy, y and x, according to the 
rule established in article (68). 


Performing these operations, we have 


yrd?y — ee yy? ~aidyde—adeiy Qrdx? = 


which, by reduction, becomes © Pee is bs 4 
(y?— a) dy + Qydly? —Sadzdy-+ Sedat = a. 


This is the second differential of the proposed equation ; ; and if we 
combine it with the first differential, we can eliminate dy, and the 
result will give the expression of dy, in terms of a, dx, and y. 
From this the function y ih be eliminated, if required, by means 
of the proposed equation. 

Dividing equation (3) by de’ it takes the form 


ay 


Leen dy . ad od 
Putting, instead of i) its value = deduced from equation 


(2), we have “ly » 


y ax) t+ 24(—— =) — = 


y?—ax 


= fas =, 


THE DIFFERENTIAL CALCULUS, 59 


and, reducing to a common denominator 


(y? —ax) yt — 6an'y? 4-204 yh 2c ry = 0; 


but the quantity ~ 
Qny/ — 6ax7y?--22'y, 
is the same with 
xy (y?—- 3axcy+-2x"), 
which is equal to zero, by the original equation, and consequently 


we have 


d2 
(yan) 2 + 2a'xy = Q, 


By differentiating equation (3), relative to d’y, dy, yand. x, we might 
derive the third differential of the proposed equation, and by elimi- 
nating d’y and dy, by means of equations (2) and (3), we might 
deduce the value of d’y; dividing this result by dx’, we should 


have the expression for the coefficient oh By continuing this 
process we should arrive at the differential coefficients of any order 
whatever. 

66. In order to give a general expression of the differential of 
the equation f(z, ¥) = 0, let f(x, y) be represented by u ; we shall 
have, by differentiating that function with respect to x, the term 


lc 


and by differentiating with respect to y, we shall have the second 
term | 


so that d.f(x, y), or 
du = wit dy ; 


but if y be-considered as a function of a, we shall have, by differen- 
tiating 


ae. » ee a 
bO THE DIFFERENTIAL CALCULUS. 


a ol & 


Te dy = . dx 3 


Substituting this value in the above equation, we shall find’ 


du = de +e oe at ye 


‘ 
From what has been demonstrated, afuele (27) we see that, u be- 
ing considered asa function of y, and y as.a function of z, the 
product ; ; 
du dy a ne 
aos € oe 
is nothing more than the. differential of u taken tbe to x 
which is comprehended in y. 
The total differential of a function of z pad y, being given by the 


equation ‘i 
= du du 
vl fe 


the expressions 


ty 


du i * 
Fede and ay, 


are called the partial differentials of uw. ad 
Inlike manner, if u is a function of three independent variables, 


Ly Yy Z, we anal have 
eee du du *du | 


du Bough Bh sin sana. 
and the terms “de, Tait 5; dz, are the partial differentials of w. 
It is proper to remark, that when u depends on two or more va- 


riables, we must not confound - dz with du, which it might be if 
f vc i . 


u contained only the variable 2, because the expression ae has, in 


the case of u being a function of two variables a particular mean- 
ing ; it denotes the es coefficient taken on, the hypothesis 


of x only being variable ; and i signifies the same with respect 
toy. The consideration af Ribbons depending on two or more in- 


dependent variables shall be resumed in a subsequent section. 


a 


hy 


THE DIFFERENTIAL CALCULUS. 61 


ry 


o7. It im be observed here that, having found the difterential 
coeflicient, a Y, oflthe equation f(x, y) = 0, ee y as an im- 


plicit function of x, we can readily find the Misiential coefficient, 
%» 


ry, of the same equation, regarding x as an implicit function of y. 


For, the differential of the equation is equal to zero, whether y 
be considered an 1 implicit ot of x, or x an implicit function of 


y 3 and, since the value ars is found from the first differential 


equation, it i Evident that the value of ie may be found from the 
same, and that -. value must be the eee, of the former ; 


dx ] 
that i <I = ———, 
is to ny. it (2, y)3 then — F a m 
verified by, the equations discussed in articles (62) and (65). 

68. The observation in article (20), respecting the constants 
which disappear by the differentiation of functions, is equally appli- 
cable to equations. If we had, for example, the equation y? = ax 
-+-b, its differential 2ydy = adzx, being independent of b, would 
equally belong to all those equations, which result from giving to 6, 
all possible values in the proposed equation. 

But we might also in the present case arrive at an equation in- 
dependent of a, although the differentiation has not made this con- 
stant disappear; for this purpose it is sufficient to eliminate a be- 
‘tween the two equations 


This may be 


= Ga+b, and 2y/y = adz, 
dhd.wé’should find’ | 
ydx = aydy-+-bda. 


Although this latter equation is not the direct differential of the pro- 
posed one ; it is, however, derived from it in such a manner, that, 
being divided by dz, it expresses the relation which ought to sub- 
sist between the variable z, the function y, and the differential co- 

i " 
efficient”, whatever be the value of a. 

If the constant which we eliminate is above the first degree in 
the given equation, the result at which we arrive will contain higher 
powers than the first of dy and dz. As an example, let us take the 
equation 


ye — Zayas? = a’. 


“és 


at 


4 
62 THE DIFFERENTIAL CALCULUS, 


By differentiating, we have | + 
ae Ft *, , 
whence Ma . Bie, 4 

a =+—+—_ _-, a ‘ 
dy 7 
and substituting this in the proposed equation, after having arrang- 


ed the result, according to powers of dy, and having ‘divided the 
whole by dx?, we shall obtain 


(x?—2y? i ~42y we = =O: | 


such is the relation which exists between the aright rel the func-— 


tion y, and its differential coefficient a imi | of any par-’ 


bg ; we 
ticular value of the constant a. i ‘ . 


By solving the equation ‘ie ll 7" 
yf —2ay-+-x? =a, st = 


with respect to a, we deduce y 
a= ~yt/ (2442), 
and the dasha a, being now disengaged from the variables : « and 


y, differentiation alone il be sufficient to make it disappear. 
Thus we arrive at 


4 29g ty Bo. 


pa J y+ ame 
By clearing this equation from the radical, we may convince our- 
selves that it is the same as that which results from elimination. 
69. Any number of constants may be made to disappear by dif- 
ferentiating as many times as there are constants in the equation. 
Let 7? = ealae( a" — 2"), 


. we have ydy = — Ais : 
differentiating again, we find 


yd°y+-dy?= — mdz’, 


s 


and substituting for m its value =yty 


ar? derived from the preceding 


equation, it becomes 


=) 


FY ' : “bp 


THE DIFFERENTIAL CALCULUS. "63 
wen ' 7 . 


Pe? 2 pa 
Va ~ dz? ey Be 
a result independent of the cotstanii tina ed 
, 16. Differentiation, combined with stil affords a means 
“of ce aS disappear from an equation : take, » 
for examp 


ar e\ 4 oye ” “50° F ha, x 


‘ar 


ae is an ndays, % | 
# A ae or 

(@-88)" 2 Quede pe 
et 
FS cy nm et li 
owe . by substiiting y y ee (ax)", yand ates of fractions, we have | 


ail on(@+a%) dy = 2myede, © 


an equation hich free from radical quantities. | 

We would have ‘arrived at the same result by iaking the tae 
rithms of ech aide ‘of the proposed equation, by which means 
we obta lad 


i 


ly = <I. (a?-+2%), 


and differentiating 


SP. me Aad 
ve m9 Wasaga ; 
and consequently, ee ab )dy = 2myxdz. 

71. We shall conclude this section with a few practical exam- 
ples in the differentiation of equations of two variables, and the eli- 
- mination of their constants. 

Ex. 1. Given the equation 


y? — Amaya" — a* =) Oe 


in which y is an inp ieae function of x ; the first differential coefti- 
cient is 


ay 
THE DIFFRREN TIAL oe 


Bx. 2. Gren the equation, | 


64a 


Pa2y(e— a)/(@—b) +(e ~a)(e=b) = 0, 


in which 3 y is an implicit function of x ; 5, the first differential coeffi- 
cient is Se . 


Ex. 3. Let the primitive equation be . 


y-pax-bey/(1-ba?) = ce =O; att rade 


the differential equation of the first order, atising fiom n 
tion of. a, is 


$y dye 4 a 
wat | are : Saat aa ise 
Ex. 4, Niet the “tof a eqbiasioe be 
= 2ay — eb = — 
the elimination of a gives ~% : i - 
dy dy 
— reas | a oe YJ —— =e 4) 20 3 
dV Vg? hee J hs 


; ius as 
and thatef 5. Soy , 


dy £4 fe 
C—O pail == O. ‘ ° ° ° * e @ (2). 
dx 


Again, by eliminating 6 from (1), or a from (2), we get the differen- 
tial equation of the seubod order 


SECTION VIil. 


Maxima and Minima of functions of one variable. 


72. Let u be-a function of the variable x, and let three values of 
“ corresponding to « = 4—1, 2 = a, x = a-+2, be 


u = f(a—1), 
u’ = f(a), 
uw” = f(a-+r). 


wr. 


» , 
THE DIFFERENTIAL CALCULUS. 65 


if @ be such a value of «, that for any finite value of-¢, however 
small, the quantities uw’ —w and u"—w” ‘have the same sign, and 
continue to have that sign for all values of 7 between that finite va- 
jue and 0, then the value w” is called a maximum or minimum va- 
lue of the function, according as the common sign of the quanti- 
ties u” —u' and u'—u'"" is plus or minus. 

73. From this, which is a rigorous definition of maxima and mini- 
ma, as Larpner properly observes, it will be perceived that these 
terms do not necessarily signify the greatest or least value of the 
function. It is true thatif the function is incapable of unlimited 
increase or decrease, and therefore has a greatest or least value, 
this value must be a maximum or minimum ; ; and this case will be 
found to come within the preceding definition. But on the other 
hand, the function may have maxima and minima values which are 
not its greatest or least values, and may even have several maxima, 
and several minima of different values. This will be easily con- 
ceived, if, while the variable x is supposed continually to increase 
from 0 to infinity, the function be supposed to vary, and in its va- 
riation, alternately to increase and decrease, the value of the func- 
tion, which stands exactly between its increase and decrease, ‘or at 
which it-changes from its increasing state to its decreasing state is 
a maximum, and that value at which it changes from a decreasing 
state to an increasing state is a minimum. 

4. Take, for example, the function u = 6=(x—a)*. In this 
equation, if # = 0, u = 6-a"; if b be supposed greater than a’, 
this value of wis positive. As 2 increases from z = 0to2 =a, 
the quantity (c—a)? diminishes from a? to0, and therefore wu in- 
creases from u = 6—«a’? tou = 0b. When = is greater than a, the 
quantity (x—a)’ again increases, and therefore u diminishes, and 
therefore the value u = 6 stands between the increase and de- 
crease of the function, and is therefore a maximum. 

This may be easily verified by substituting successively « ~ 7, a, 
and a-+7 instead of x in the proposed function, we find 


u == b 17, 
ma’ = t, 
U = bm; 
therefore, wu" — wu = 6—(b—7) =, and ul! —w" = b—~(b— f= 7, 


Again, letu= b-+(x—a)*. In this case, when x = 0, u = 6--a?, 
the quantity (x—«a)° decreases from « = 0, to x = a, for which 
u=b, When « becomes greater than a, wu begins to increase : 
hence, in this case, u == 4 is a minimum value of the function. 

K 


* 
66 THE DIFFERENTIAL CALCULUS. 


‘This may be easily verified by substituting Soe Gants By 
and a-+-2 instead of 2; we find w = 6+:7, Ui Ge = b,and wu” = 6-+ 
; therefore, u’—w' = —23, and u’—w!” = —? 5 consequently, 
since the values of u’—w and u’—wu” are both negative, b is the 
minimum value of the proposed function. 

Every function which increases or decreases continually, whilst 
the variable on which it depends 1 increases, admits neither of a 
maximum nor a minimum, since each sugeeemin ge value is always 
greater or less than the preceding. 

The essential characteristic of a maximum consists in its being 
see than both the values which immediately precede and follow 

; that of a minimum, on the contrary, in its being | less than both 
wa values. apt 

We will now proceed to state the method of discovering the 
maxima and minima of functions of one variable. _ 

75. Let uv’ be any function whatever of 2, in wk nich the variable 
has attained that value which makes it a maximum or minimum : 
from what has been already observed, it follows, that if we exa- 
mine the values of u’ = fx, corresponding to x—2, and «-+1, we 
ought, (however small the quantity 2 may be), to obtain results less 
than the maximum, or greater than the minimum. Denoting by 
uw the value of wv’ which corresponds to «—2, and by w’’ that which 
corresponds to «-+/, we shall have by Taylor’s theexsyp (53). 


w= ft — Peekp" 4 +--+, &e. 
w = fe 


= fe-bp’. “tp ep v obs &e. 


a 


P>P>p , &ec. expressing the successive differential coefficients 
of the function ; therefore, by subtracting successively the first 
and third equations from the second, we shall have 


ul —w =i pp tp MT ieee, : 


es f ” 2 er Ji 
usu =. er ed doer mae o 5.3. ° ° e 


Such a value may be assigned to z as will render the term p’ great- 
er than the sum of those which follow it (11) ; and, since this term 
enters into the values of «’—w' and u’—w'” with different signs, it 
follows that one of these quantities must be greater than ’, and the 


* 
, THE DIFFERENVIAL CALCULUS. ~ 67 


other less : consequently the proposed function cannot be a maxi- 
mum or minimum unless p’ be equal to nothing, and as p’ is a func- 
tion of a, it follows that no value of x but such as are roots of the 
equation p’ = 0, can render the function either a maximum or mini- 
mum. 

Therefore, if the first differential coefficient vanishes, we*'must 
have, i in that case 


” mn a | 
wad = 2 3p'tp gf RNS se t. 
i: ak ee 72 —p" —p v 
j DS PM ea A 
dy 2 


in this case, as before, such a value may be assigned to 7 as will 
render the ee term greater than the remainder of each series. 
Hence the quantities w’—w’ and wu” —w'" will both have the sign of 
—p", and wills 1ave the same sign, for every value of 7 between that 
and 0. . The ‘corresponding value of the function will, therefore, 
be a maximum if p <0, and aminimumif p”>0. 

From this it follows, that the value of X, which ws necessary to 
make a function u a maximum or minimum, (for they are both giv- 
en by the same equation,) 7s that which, when substituted in the dif- 
ferential coefficient s will make it equal to nothing 

In the example, y = b—(«—a)’, considered in the last article, we 
have 


and putting —2(r—a) = 0, we havex = a. - To discover whether 
this value relates to a maximum or minimum, we must examine the 


2 


d 
value of 2 
dx? 


; and since thisis reduced to — 2, a negative quantity, 


we mayconclude that the value x = a gives a maximum value of wu. 
Treating the function 


’ 


“y == b-+{(r—a)? 


> dau. 
in the same manner, we find « = a ; but in this case Fay will be a 


positive quantity : this value of x, therefore, in the case before us, 
corresponds to a mnimum. 


du : 
76. We must not, however, conclude, because aie 0, in the 


65 THE DIFFERENTIAL CALCULUS. 


case of a maximum or minimum that the one or the, other must 
necessarily take place whenever this condition is, fulfilled. In fact, 
if the value of « which makes p’ = 0, causes, at the same time 
p’ = 0, and not p’’, the series become 


id ’ “ +t veye 2 
ual =P} tp apt agrary . 8 ie 


be § / ” t 
UmmY = 8 soe 9 Oa!” —_— A . - 
a FOR eat ey Pr 


In this case, as in the first, a value may be assigned to 2 such, that 
uw" ~w'shall have the sign of +p", and u’ —u”" the sign of —p’’, and 
therefore the corresponding value of the aor is not either a 
maximum or minimum. o 

For instance, let u = bran 5 ; we have by successive diffe- 
rentiation | Py, 


TF 
iy 


24 
—- = 3(4#— a)’, a 2x (aa), = 6. 


’ 
Therefore, if the proposed function adit of a iidesshamiannn or mini- 
mum, we shall have 3(a—a)* =.0 ; and consequently « = a : this 
value being substituted in the seer differential coefficient, gives 


7 


but the third differential coefficient does not vanish for the same 
value of «, since it is equal to 6; hence the function does not ad- 
mit of a maximum or minimum. 
if, however, the value which makes p' = 0, Pp" =.0, causes at 
the same time p”’ = 0, and not p’” = 0 ; we have 


Co ae ; aeeaoce 2 : f 


Nth 


ay lar aay a3 ; —p"" —p"! A: tee t i ¢ 
2.3.4. 6 


in which the conditiGi relative to the maximum and minimum 
would be again fulfilled: and'we may discover from the sign of 
p’” which of the two takes place : for the function is a maximum 
or minimum, according as p”” <0, or >0; and so on. 

In this manner we shall find that the value « = a gives a mazi- 
mum for the function y = b—-(a—2x)*; for, by successive differen- 
tiation, we shall find 


aus 


THE DIFFERENTIAL CALCULUS, Ho 


== — 24; 


Putting the first differential coefficient equal to zero, we find « = a ; 
this value of x makes the second and third differential coefficients 
also equal to nothing, but the fourth differential coefficient does not 
vanish, and its value has anegativesign; we must therefore conclude 
that the value z = a gives a maximum for the proposed function. 

By proceeding in like manner we shall find that the value z =a 
gives a minimum for the function y = b-+-(a— <x)’. 

77. Hence we conclude, that in order to determine the maxima 
and. minima values of a function, it is necessary first to find the first 
differential coefficient p’. This being, in general, a function of x, de- 
termines those values of x which render it = 0, or the roots of the 
equation p' = 0. No values of the variable x can render the func- 
tion « a maximum or minimum, but such as are found among the real 
roots of this equation.* Substitute these roots successively for x 
in the second differential coefficient p’. Such of them as render 
p < 0 being substituted for x in the function u give maximum va- 
lues ; such asrender p” > 0, give minimum values. If, however, 
any of them render p” = 0, they must be substituted in p” ; and if 
they render it > or < 0, they will not render the function u either 
maximum or minimum ; but if they also render p” = 0, they must 
be substituted in p””, and so on: andif the first differential coeffi- 
cient, which they render > or < 0, be of an add order, they do not 
give cither maxtna or minima. ; but if it be of an even order, they de- 
termine maxima or minima according us they render that differen- 
tial coefficient negative or positive. 


* There are, however, exceptions to this, since the first differential coefficient be- 
comes in some cases infinite, for particular values of the variablew; and at the same 
time the proposed function may admit of a maximum or minimum. For instance, let 


2 
u=b.c(z—a)° : it. is obvious that when =a, this function is a maximum or 
minimum, according as c is positive or negative ; a case in which we have 
du 2 ( ) vf 

—— = —-(a@—a)— F = —. 

dx 3 i: 
Let V = (u—b)? = c3 (a—a)?, an expression which is a minimum or a maximum 
at the same time with vw: making therefore. 


— =n 2¢7 (~—ay = 0. 


we fiad, when x = a, V =3 9, or U=sb, aminimum or a maximum. according to the 
sign of ¢. 


, 


TO THE DIFFERENTIAL CALCULUS. 


78. Let us now consider the example, u == ax’ —ba?-+-2-+-9. 


d | 
Hence — = 3ax?——2ba-+- 1. 
dx 


The values of x which render this = 0, are 


_ b+4/(b?—32) 
By Wye? 
ba / (tir BON ge 
so VO. 


If b? < 3a, these values are both impossible, and therefore the 
function in this case is not capable of a maximum or minimum*, 
But if 6? be not < 3a, let the function be differentiated again, and 
the result is 


substituting in this the value of x already found, 


d2u 
dn? <= oo Hh 2,/ (0? —3a). 


If 6? > 3a, one value of x MN 5 ae > 0, and the other < 0. 


Hence in this case, if 
b— /(b?—3a) 
Cy Me. Vea 


be substituted for x in the function u, the corresponding value will 
bea maximum ; and if 


b-++./ (b°-—3a) 
3a 


be substituted, the corresponding value is a minimum. 
If, however, 6” = 3a, and .*, b°—3a = 0; in this case, the va- 


dt b 
lue of « determined by A = 0is ct which being substituted forz in 


| ae 
Nea —— 5 7 Gax—26, 


*Ttis hardly necessary to remark, that no maximum or minimum value of w can 


: m j _ du 
ever arise from an imaginary root of the equation 7— = 0. 
~inary FE 


THE DIFFERENTIAL CALCULUS. 7k 


a2; ‘ ; 
renders = = 0. It will therefore be necessary to differentiate 


again, which gives 


This not depending on x, and not being = 0, the function admits 
of no maximum or minimum in this case. 


; 1 
79. [tis proper to observe, thatif u be a maximum, r must be 


a minimum, and vice versa : this consideration will sometimes ena- 
ble us to shorten the necessary operations : for example, let 


By differentiation we find that, if « = 1, 4 = 1, amaximum ; and if 
2=—1,u= —},a minimum. But, if instead of differentiating 
the proposed oe we take its reciprocal, the differential oder. 


ficients of | — or ot, are more readily found than those of uw. Let 


1 ; F ne : ; ames ; 
— =u, or uu = 1; now, by successive differentiation, we obtain 
u 
du’ i Pi g@Msey 2 
dx 2? dx? 43 
~ du! ay 
iti a i ¢ =a $e a aD 

The condition re 0, givese = 1, « ToS +2, Hence, 


ifas1ls) ou = 2, a minimum, and vu =! a maximum: if x 


=—1, .. 4» = —2,a maximum, and vu = —1, a minimum. 
80. In finding the maximum and minimum values of functions 
under a fractional iu it will frequently be useful to remark : 


ist. Thatif « = — 5» both being functions of x, we have du =: =, 
d dv’ du ; 
es Asia ; éondeguadliy) when — = 0, we must have either 
v 7) dx 
dv dv ; 
ve) = 0, —_ ol 9 or Vv = Ce. 
v vr) 


) d2 d2v' 
Qdly. Also, du = du ee ee ; ps Sia _ 


v 
shel at 


a et ee 


4 


, when du = 0; and since v” is essentially positive, 


oa} 
i 


THE DIFFERENTIAL CALCULUS. 


2 


d 
the sign of 5 will depend upon v'd?v—vd?v', when we substitute 


for x the roots of da =e); 
da 


| on wm §ndv — adv 
3dly. If u = -—-, we have du = —— ) — —- ; and 
yr ? ~' ee het 


fo v 
du ndv mdv ee 
and therefore, Sour 0, when vy” = 0, — = —-, orv’” = @. 
v 7) 


Athly. Also, when * = 0, we find ~ 


ee ee ee 


nd» nid’y  ndv (f ah f 


v v v 


from which the sign of d’u may be easily determined. 


3 3 
Let, for example, u = “= a 
i Lacey la Pag ne 3d. (3d.(e+3)* 
When, tot =0; we have vw («+3)? = rr ~Eee3y 
QO\2 
aS —; from the first of these two equations we find x = 
H 0 


a 2 
—-3, and from the second x = 0: now, since the sign of me = 
will depend upon vd?y—vi'y'! or (x-+2)*.6(a-+3)da?—2(2-+3)° 
dz*; .*. by substituting successively —3 and 0, for x in (#+-2)?. 
6(z-+3)—2(x-+3)*, we shall find inthe first case, its value =0; 
and in the second, its value = 18. 


ee 27 beh i, . . 
Hence, if « = 0,u = 7 2 minimum ; and if x= —3, wu is 


neither a max. nor mon. 

This function becomes a maximum when x = —2, a case in 
which wu as well as its differential coefficients become infinitely 
great; this case can hardly be an exception to the general proposi- 
tion in Art. 77, the trut: of which entirely depends on the differen- 
tial coefficients of the function being finite or evanescent, upon the 
substitution of a particular value of +. may be explained as 


v r— x)" 
follows: let wu = -———-——, and therefere” ah” Sa )"Q a quan- 
a 


(a= ayrQ . 


tity which may be readily shown to be a minimum when « = 4, if 


q Hes I 
n be an even number; and since the minimum values of — are the 
Ut 


W 


{THE DIFFERRNTIAL CALCULUS. 73% 


mea. Values of u, it is obvious that in this case we must reckon wu 
“4 ; ; “ Seri 
=, among the maximum values of the function. 
81. It may be observed here, that,in general, ifany function u be 


oe ‘ " m * 
a maximum or a minimum, we likewise consider u" as a mazi- 
‘mum or minimum, unless m be an odd and » an even number ; in 
which case it is necessary that the maximum and minimum values 
of u should have a positive sign. | 
In finding, however, the maximum and minimum values of u 
from those of u™, there may exist minimum values of the second 
function, which have no such values i in the first; for the maziina 


and pane? of w™ are determined equally from the equation u™—! 
= 0, and SY = = 0; though no such value arises from first of 


these ase unless m be an even number. 
But this statement requires some modification; for if V = uw, 
adv u 
it se not always follow het fa = 0, when u™—! = 0), or ihe 0, 
since he same valuc of x, attra verifies the first equation, may 
‘i 


#3 


w dV 
make pi == ©, and vice versa 5 it ‘appears: therefore, that “Te MAY 
dx 


have a finite value under these circumstances, which is inconsistent 
with the condition of a maximum or minimum value of V. 


Let, for example, 1 == taf (ae—2"). /~ 
3a 
By differentiation, we find that if « Tia) 


ey 


3Vv¥3 


b= iar a, a maximum. eee 


Assume V = u*, from which we get 


; oe : ' 83a . 
The roots of the equation Bax’ —~ 40° = 0, are v% 0,0 ; the first 
of which makes V a maximum ; but v= 0 when x = a, a value 
dV | 
which makes ee = o,and ig a finite quantity. ( 
act 


Again, if we take the equation u = x(%—a)*, we shall find, if 


gg 3 270.4 
2S = 
4. ; 


, a maximum. 


i\ 


L 


‘® '74 THE DIFFERENTIAL CALCULUS. 


If we make V = u’ = 2°(a —a)°, we find 


AV d 
mae #2 Qu = 2x(e— a)? (40a) = 0: 
wh 26 
consequently, if = -, y= ig a®, a maximum: xa 


ifier=—av = 0,2 eS 2 ee 
If «= 0, eer amin. my) 
In this case u? admits of two minimum values which have no ex- 
istence in w. 3 . 
82. It may not be improper to discuss a few more cue we : 
fore we proceed to the solution of problems in Geometry, which 
involve the consideration of maxima and minima. : 
Ex. 1. To find the number which bears the least ratio to its lo- 
-garithm ; let « = the number, then 


x 


ue ——5 


Ix’ 


by differentiation, w we find that if lx = 1, or, ane amounts to the 
same thing, if «=e, the base of th shin system, we 
would have =e,a minimum. a 

Exy2. To find — number 2, whose zth root is the greatest 


possible: or let u = 2, yg 
~ Taking the logarithms, we have 


1 4 wa 
lu= : lx ; .*, by differentiation, 


du 


ie, 
dx ei eli 0; 


and consequently, lz = 1, or x = e. 
i * 
- Hence u = e* » a maximum. ce 


Ex. 3. “To divide'a number a Mito two parts such, that the pro- 
duct of the mth power of one, and the nth power of! the other, shall 
be @ maximum or miniinum. 

If, x be one of the parts, and .°. a—z the other, the product is 


’ oF z™(a- =©)”, a | & 
du 


= (ama) 2m, mamta) 


hy 


HE DIFFERENTIAL CALCULUS. 


~] 


or 


a2 
“= = (A—xr)"—2 9" §(ma— (m-bn)x)? —m(a—z)* ~na*t. 


d 7 
The values of x, which render “- = 0, are determined by the equa- 


tions 
Hf ~... . 
a—x =0,x =0, ma—(m-+n)x = 0, 
eaive 
a ma 
gr=sa,r=—0,¢= 
m+n 


The value « =a renders the second differential coefficient = 0; 
and it is evident that since every differential coefficient of an infe- 
rior order to the nth will have x—a, or some power of it as a fac- 
tor, the same value of zx will render all these = 0. . The nth diffe- 
rential coefficient will not have the factor «—a, and therefore, in 
it changing z into a, the result will not be found = 0. If n be 
odd, therefore, this value of x does not correspond to either a max- 

‘imum or minimum; and if » be even, it will be found that the va- 

lue x = a renders the nth differential coefficient > 0, and that 

therefore the function is a minimum. 
Similar observations apply to the value x = 0, by considering 
a—Oas a factor of the differential coefficients. The value 2 = 


d?x 
being substituted for x in ——, renders it negative, and there- 


re dx? 

fore renders the function a maximum. We shall now proceed to 
the application of the theory of Maxima and Minima to the solu- 
tion of Geometrical problems. 


Prosiem. I. 


83. To determine the greatest rectangle that can be inscribed in 
a given triangle. 3 


Let, Fig. 1. 4D =e. BC =a, andAP = x: then, because 
the lines BC, MM are parallel, we have 


a:e+: +: = MM’; therefore, vu = MMN'N = <(e—2) i is a 
a 


é : : ‘ ° 
maximum, when x = =, or when the altitude of the triangle is bi- 


sected. 


76 THE DIFFERENTIAL CALCULUS. 


PRoBLEM IT. : 


(84. To find the greatest right angled triangle that can be con- 
st#ueted on a given right line. 


Ilet the given line 4B =a, Fig. 2, and «.= BC, one of the 


sides of the triangle, the other side 1C shall be represented by 


»/(a?—x*); and consequently the area of the triangle will be ex- 
pressed by sit - 


er Ne eae 
Be 
5 


ov (a2) 5 


thus the equation ar the oe seu be, 


= ave x" 2), oru = say (i By art) 5 


whence we deduldes by. giibrenttatiol; 
du of | atx—2e9 aa a % “e gee 20). 
dx 2af (aia*— 2") ~ 2y/ (a2 a3 242 — x4) 

Yiuiiing this value equal to zero, we have | 


nus OF ear ON Or 2 (828) = 0; 


an equation from which we derive 
a . ih 


gs! 0 or 2r7 = a. 


Asa cannot be nothing, its value must be determined from the se- | 


cond.equation ; this value shows that the two sides “ the triangle 
are equal to each other. 
Now, since a2" = 0, by Hypothesis, we have — (a? —2x?),. 
wae ay neat nie Tan = 0; and therefore the second. differential 


coefficient of the proposed equation, gives 


d2u phe x d.(a?--27) os 2x2 y@ ay, 
dx? 2.4/(a3 a? —x1) dx mes / (02x? —x4) ’ 
and substituting ta? for x? in this result, we shall find 
d2 x i ae Qa2 
—— ee =. ome ended) Anata aha ee ote 
da? $7 (2at at) a? 


This result being negative, the hypothesis of a2? = 0, deter- 
mines, for x, a value which corresponds toa maximum. 


THE DIFFERENTIAL CALCULUS. var 


+4 


Pros.yem Il. 


85. To determine the greatest cylinder that can be inscribed in u 
given right cone. ‘ . 


: 


Let a, (Fig. 3), be the height SC of the “pe, b the radius AC 
of its base, and x the distance SD from the summit to the centre of 
the circle EGF. 

‘The similar triangles SAC, SED give 


SC: AC :: SD: ED, 
greet aibst2:ED; 


Let 1: be the ratio of the diameter to the circumference 3; we 
know that the circle, whose radius is 7, has for its area rr’. 


Therefore the area of the circle EGF. which has “3 for its radius, 
(2 va il 

is — ; multiplying this area by the height DC of the cylinder, 

that is to say by a—x, we shall have 


wb2 | 
ay x*(a— x) 


for the solid content ‘of the cylinder ; thus the equation to differen- 
tiate is 
_ @b3 a 
= ye = . . 
a = (a a 
we deduce from this equation 


du «63 du xb? 
oe = = (@ax— 322) and —— = ~--(2a—6r) 
da a? (eRe a) ame de a? arr, os) 


d 
putting the value of > equal to zero, we have 


wb? 
Dany 2) Q,.2 —— 
a (2ax —32") = 0, or 2ax—32? = 0, 


an equation which is the. product of the factors 2 and 2a—3z ; 
and consequently, 


THE DIFFERENTIAL CALCULUS. 
2a 
x =0,or 2 Si 


the value x = 0, cannot Sonapond to a maximum, since, in this 


fy Qarb2 


case, “ becomes oom Potaitive pombe 


da? 


lk ne 
This value of  Mindicates a minimum, in fact, when z = 0, the 


: ae : ‘ ‘ 20. 
cylinder coincides with the axis of the cone. The value x a oa 


therefore the only one which answers the condition of the ques- 
vill. d? 2b? 

tion; and in this case, a= oe 
it follows that, the inscribed shies will be the greatest possible, 
when the altitude thereof is just one third of the altitude of the 


cone. 


, anegative quantity, Hence, P 


4 


Prosiem LV. 


86. To determine the dimensions of a cylindric measure, which 
shall contuin a given quantity ue water under the least internal Su- 


perficies pasts: 


Let V be the given content of the cylinder, and # = the radius 
of the base: z* shall be the area of the base: and since the 
height of the cylinder multiplied by its base is equal to its solid 
content, we shall have 


height of the cylinder Xru* = V ; 
-, heighivof ihe cylinder = —— ; 
. height of the cylinder = asi 


multiplying this height by the circumference of the base, which is 
2ax, we shall have 


2 
— X2rz = —— 


re? % “i 
for the concave surface of the cylinder. If to this surface we add 
xx", which is that of the cylinder, the equation to be differentiated 
shall be 


QV ' 
Uo ——-+9r2’, 1 Reels 
xz 


and by differentiation we shall find 


THE DIFFERENTIAL CALCULUS. 79 


du, . ia 
The value of ne being put equal to zero, gives 


This value answers to a minimum, because it renders the value of 


5 positive : if we put 2/ at for x in the expression of the height, 
m | 


_ we shall have, for the height of the cylinder 3/—-. Hence, it fol- 


lows that, the diameter of the base must be just double of the alti- 


87. Exercises in th 


€ maxima and minima of functions of one 
variable. 


Ex. 1. Let « = 2'—~S8x°--297?—240-4-12. 
| If c = 3,u = 3, a minimum, 
If «= 2,u = 4, a maximum. 
Ife=l,u= 3, a mimmum. 
Ex. 2. Let w = 25—5e2'+-523-+-1. 
If + = 3, u = — 27, a minimum. 
If «= 1, u = 2, a maximum. 
ifs=0,4=1, Whidh" is neither a maximum, nor a minimum, 


' ett 
since two roots of the equation he 0, are equal to nothing. 


£2 at I 
E ° e L t == + ry 
, et u ere ; 


pate Ori 


o£ 
° 


fe If x= 0, u = —1, a mextmem. 


a . 


If 2 = 2, m= 2,0 meni. 08 YO 


3 " 
Ex. 4. Letu = Wa: 


4 — 2+ 
If c= 1, u = 2, a maximum. 

is x 
ifem—lus —2, a mnimum, 


. 


” 


20 THE DIFFERENTIAL CALOULUS.- 


Bis ou 
The other values of x, in the equation a = 0, are 
| cs 


Wes — 544/21 
aa 2 


—1)? 
Bx, 5. = Oe 
x: Let u ery y 


9 * 
Ifo = 6) es re a maximum, 


Ifc=1,u= 0,a minimum. 


i 


If c= —1, u= co, which is neither a maximum, nor a manimun, 
142) ) 
Ex.’ 6. Létw= ( ce.. 
: (Fa) 
i 17+-7 7 
if ¢ = —2+/7,u= en , a maxImuUM. 


ah 
ifs = —2—S/7u= ee aameninum. 


There is no maximum or minimum, corresponding to x = — 1, or 
cmt f(— 1). 
Ex. 7. To divide a number-a into such a number of a) 


parts, that their continued product may be a maximum; or let x 
a 


ay a # 
ee (-) _ if «=e, u, = e ,a maximum. 


Ex. 8 Of ali rigat angled plane triangles contarning the same 
given area, to find that iter the sum of the two legs is the least pos- 
stble. 

Let one of the legs be denoted by x, and the area of the tri- 
angle by a ; then it will be found that each of the legs is equal 
to ,/2a, and consequently, the two legs are equal to each other. 

Ex. 9. To determiné the least zsosceles triangle which can cir- 
cumscribe a given circle. ‘ | ie 

Let the distance of the vertex of the triangle from the centre of 
the circle, be denoted by x, and let the remaining part of the per- 
pendicular, which is the radius of the circle, be represented’by a : 
then, we shall find x = 2a ; and therefore, when the triangle is the 
least possible, it is equilateral. 

Ex. 10. Of all cones under the same given superficies (s), to 
find that whose solidity is the greatesi. 

If p denote the periphery of the circle whose diameter is unity, 


THE DIFFERENTIAL CALCULUS. 8i 


and the semidiameter of the base = 2; then, we shall find x 
=V(z 5) and the length of the slant side = = 3uv(z, ) : from 


whence it will appear that the greatest cone, under a given surface, 
(or a given cone under the least surface,) will have the length of 
the slant side to the semidiameter of the base in the ratio of 3 
to 1, é 


SECTION IX. 


The effect of particular values of the variable upon a function, 
and its differential coefficients. 


88. A function is in general rendered either positive or negative 
by the real values which may be assigned to the variable. ‘There are, 
however, four states of the function which are attended with pecu- 
liar circumstances, and which require some examination. Certain 
particular values of 2 may render the function, or its differential co- 


efficients = 0, oe imaginary, or infinite. We shall consider 


these four cases in explicit functions, 

89. To determine the successive differential coefficients of a func- 
tion (u) which correspond to any particular value (a) of the va- 
riable (x), which renders the function or any of ats differential co- 
efficients = 0. 

In the first place, let s = a render the function itself = 0. By 
the principles of Algebraical equations, it follows, that « —a, or some 
positive power of it must bea factor of wu ; so that w must be of the 
form u = P(z—a)™, m being a positive integer or fraction, and P 
being a function of x not divisible by (2 —a), or any power of it. 

From the process of differentiation it appears that (x —a)"~", 
(a—a)"—?, &c. are factors of the successive differential coeffi- 
cients of wu. Let these coefficients be represented by wv’, u’, yu” 


we ee. wu), they must be of the forms 
m= (2 os ay, 
1 ee eer ~a\"—*, 


M 


82 THE DIFFERENTIAL CALCULUS. 


80 Oe Oe) et. a kere le 


Ur = PO (a—a)"—™, 
where P P", Xc. are quantities not divisible by any power of 
(x—a). 
If m be an integer, these successive differential coefficients will 
be = 0 when z = a as far as the (m--1)th inclusive ; but the 
mth differential coefficient will be of the form 


au) = PEY(4— aes = Pt), 


which not being divisible by any power of («—a), will not vanish 
when x =a. The same may be observed of the differential co- 
efficients which succeed the mth. For example, take the function 
a°— x? —a’x-+-a°, which vanishes by the supposition of x =a; 
its first differential vanishes on the same hypothesis, but its second 
differential, which is (6z—2a) dx? does notvanish. It is, there- 
fore, freed from the factor (z- —qa); and since two differentiations 
were hecessary for that purpose, we conclude, that it is of the 
form P(«— a), which may be easily ascertained by other means ; 
for 


— A2°—are--a? = (x-+a)(x—a)?. 


It is plain that if m = 1, the function vanishes, but none of its dif- 
ferential coefficients will vanish. For instance: let u = 3x2 
(v—-1). then, when x= 1, we have u == 0, a = 8, and au 

x dx? 
=12. If m bea fraction, let x be the next integer below it, and 
therefore n-++1 the next above it. In this case the differential co- 
efficients, as far as the mth inclusive, vanish with the function, and 
those that succeed it all become infinite. This is evident from con- 


sidering that m—n is positive, and m—({n-+1) negative. For ex- 


ample, let w = ax(x—a)?; here n = 2 the next integer below the 
exponent 5, and » = 3 the next above. It is evident that the func- 
tion vanishes, when x = a ; but, by differentiation, we shall find for 
the same value of x, the following differential coefficients. 


du d2u vit diy d4y 


w= 0 ——~'== 0, ——_ =a, &c. 
dx ” dx > da3 “det ee 


Hence it likewise follows, that if m be a proper fraction, then 
m == 0: and in this case all the differential coefficients are infinite. 


THE DIFFERENTIAL CALCULUS, 83 


Secondly. Let «= a render any proposed differential coeffi- 
cient = 0: if the first differential coefficient which it renders 
== 0, be of the nth order, it follows that 


wu) = P® (a—a)™, 


p” not being divisible by a power of xa, In this case it may 
be proved by the process already used, that when m is an integer, 
the differential coefficients from the mth to the (m-++n—~ 1)th inclu- 
sive vanish when xz = a, and those which succeed them do not. 
If m be a fraction between / and /-+1, then the differential co- 
efficients from the nth to the (n+-/)th vanish, and the succeeding 
coefficients become infinite. If m be a proper fraction, then all 


the coefficients after the nth become infinite. : 

90. Given a function which vanishes when x = a, to determine 
the highest power of (x—a), which divides the function. 

Let u =/(x), which vanishes when « =a. It is measured by 
(x—a)* to determine z. Let the function be differentiated until a 
differential - coefficient wu) be found which does not vanish when « 
=a. This coefficient will be either finite or infinite. If it be 
finite the value of zis aninteger, and =n. [If it be infinite, the 
value of z isa fraction whose value is between the integers # and 
n—l1l. ‘To determine it, let u- be such a fractional power k of 

—1 
a-—«, that the quote eae may be finite when z =a ; then the 
exponent of the sought power will be n-++k : this is manifest from 
the last article. 
91. To determine the true value of a function which a particular 


0 ° . 
value of x renders 5 or infinite. 


That the first may take place, it is necessary that the numera- 
tor and denominator be both functions of x, which vanish when =z 
== a, and which therefore have factors of the form (x —a):. 

Let the function then be 


and let the highest power of (x~a) which divides one be 2, and 
the other 2’. The function may therefore be expressed thus, 


34 THE DIFFERENTIAL CALCULUS. 


The values of z and 2’ are to be determined as in the last proposi- 
tion. 
He> 2’,u=0. If.2 2 2, is ininte. I 2-7, 


Hence it appears that the method of proceeding to determine 
the value of the function is, to differentiate both the numerator and 
denominator until a differential coefficient of each be found, which 
does not vanish whenx ==a. luet this coefficient be of the nth 
order for the numerator, and of the mth for the denominator : 
P (x—a)"—™ 0 

5: 
If n < m, the function is infinite, as well as allits differential co- 
ra =5 =o. If n =m, the nth 
differential coefficient in each term may be either finite or infinite : 
this presents four cases. First :—If it be infinite in the numera- 
tor, and finite in the denominator ; in this case z is a fraction less 
than n and z’ =n; hence the function is infinite. 

Secondly. If it be infinite in the denominator, and finite in the 
nunrerator, then z = n, and 2’ is” a fraction less than »; therefore 
the value of the function is 0. . 

Thirdly. If both be finite ; in this case the value of the func- 
tion is a fraction whose numerator and denominator are the diffe- 
rential coefficients themselves. For let #-+2 be substituted for x 
in both numerator and denominator, and the results developed, we 


find ; 


then, if x > m, the functionis = 0; for vu = 


efficients ; for u= 


: PC . 
fet dat de. 


US it a: aralaae 
e-+- B+ B’—--+ .... 
ai, rt 1.2 ¥ 


where .1, 4’, &c. B’, B’, &c. are the successive differential coef- 
ficients. Substituting a for x, the functions and their successive dif- 
ferential coefficients vanish as far as the nth differential coefficient, 
which is by hypothesis finite in both numerator and denominator. 
Hence the function becomes 


7 Mri am 
AC), a en ae, Se ee 
Bi Be 1.2 e e e 2 ee e n-+1 
Fy q gut loe RCS 


Bo), 
1.2 


at © gq @ 


er Ber"), re eee 
ri 1.2. 2-1 


ar A 


THE DIFFERENTIAL CALCULUS. 85 


Dividing both terms by 2”, and supposing 7 = 0, we find 


AM) 
BO” 


“iL 


which is a fraction whose numerator and denominator are the first 
differential coefficients which remain finite when x = a. 


— 2a a Q 
al a iid becomes u = —, when 
ba? —2bcex-+be? 0 


z=c; by differentiating the numerator and denominator, we find 
for the first differential coefficients ax —ac and 6x -— bc, or, which 


For example ; let u = 


amounts to the same thing, the fraction = — — a result which be- 


0 ' bie, Anns 
comes also 0 when z = c ; but by differentiating again, it becomes 


: therefore the true value of the proposed function, when x = c¢, 


"S18 


m8 


Fourthly. If the first differential coefficient which does not va- 
nish be of the same (nth) order, and both become infinite when x 
=a. In this case z and z’ are both fractions between the integers 
n—1andn. Thevalues of the fractions may be determined as in 
(90) ; and if they be equal, both terms of the fraction being divid- 
ed by the common power of x—a, the result will be its true value. 

Or the value may be determined thus. In both numerator and 
denominator let x2 be substituted for x, and the results expanded 
according to increasing powers of 7 by the ordinary rules of Alge- 
bra; forin this case the series of Taylor will not apply ; and let 
the result be 


Aiet- Ai Arie” os 

Bie+ Bu’ + Bi” ss 
The exponents a, a’, a’, &c. 5, b’, 6", &c. being arranged in an in- 
creasing order. 


Distinguishing the three cases of a > 6,a = b, anda < 6, we 
may, in the first two cases, write the expression thus : 


Ait Aid fia’ —b 


a? ~~ B+ Bie Be” 


36 THE DIFFERENTIAL CALCULUS. 


Under this form itis easy to perceive, that as long as a is greater 
than 6, the supposition of 2 == 0 will make the fraction equal to 
nothing, and that it will be tadyood tou = ee when a= J, since 
in that case 77> =t. If a < b, the fraction becomes 

ee ce aN, i 


Biba Bia Bee” 
making + =a andi = 0, this becomes infinite. In all cases, the 


true value depends on the first term alone of each series. 
92, Hence, the following rule extends to all functions which can 


: 0 ae 
present themselves under the indeterminate form 7% Take the first 


term of each of the sertes which express the development of the nu- 
merator and denominator, when x = a+i; reduce the resulting 
fraction to tts most simple form, and then makei = 0, the result 
thus obtained will be the value of the proposed fraction when x = a. 
If we had, for example, 


8 5 aS 
Lee) 


Ww OME aries ; 
(2—a)* 


Substituting a+7 for x, we have 


3 
tat Bare y* 


(i). Wr) 


== (Qa-}1)2; 


making 7 = 0, we find 
i VLA soar aaa 
The value may easily be found in this case, by division, since 


3 
2 


3 

(a? —a?)? L? am 1" 2 
ee | ( 4 = (r-+a)*, 
(x +a)? mr 


which becomes, when « = a, 


¢ 
THE DIFFERENTIAL CALCULUS. 87 


may become infinite at the same time for a particular value of «x ; 


:, 0 
still it may be reduced to the form 7% forit may be written thus ; 


ete mia Bi 
which 1s reduced to the form j when fx and Fx are infinite. 


If in the product of two functions of «x, one factor become infi- 
nite when « = a, and the other 0, it can be reduced to the form 


is and therefore its value may be found by the preceding rules : 


Let u = fa X Fx, and let Fx be infinite and fe = 0 when 
© =a. 


| 1 
If we suppose f'x = Fe then if =a, fx =0. But u= 


fx 


é 0 
<. which becomes — when z = a. 


Es 0 
ifu= ti , and x = a, render both the numerator and denomi- 


nator infinite, the value of w may be found by the same rules : for, 
x 


ee l . 0 
aed fe? and oe of PI dd ‘oa which becomes a 
when x =a. 

Also, if w= fa—Fx, and these functions become infinite when 


x == a; let 


1 1 
fx fe and Fz ice 3 
we Ca P’e—f'r 
ee U eft SX Pry’ 
which become 0° when « = a. Hence all combinations of func- 


tions of x coming under the preceding forms are regulated by the 
rules already delivered. 

94. We shall now proceed to give some examples of the appli- 
cation of these rules. ” 


A 


&8 THE DIFFERENTIAL CALCULUS. 


ae 20 8 
Ex. 1. Let u = Gra (4 ‘an en find its value when x = 0. 
3 72 


By differentiating © 
Be ys 
iy, (a?—2?)’ 
rr d. Faz 


d.2 fx 1 2 
de aaron 
(a? —x?)\? a 
2 T'> m 
d. Fe 9, 
dx? 


which gives, when x = 0, 
i fa 
dx? 


d2.Fx 


dx? ad 


1 
“. U = —, when x = 0. 


2a 
es 1 — sinz +- cos x rg 
Ex. 2. Letu = ———- ———, to find u whens = -. 
sinz-+- cos z—1 2 


By differentiating, 


d.fx P 
—— == — cos —sin x 
dx : 


dF 


= COS £—SIN x. 
adr 


wv 
Hence, when x = ye 1. 


GZ mm G7 


Ex. 3. Letu = , to find wu when x = 0, 


ar r 


= 67 la—belb, jal 


d.fax 
sued 


i 


a 


aor 


+h * 


®, 


te : nA 
Ex. 4. Let u= ve pf (Or). , to find. he value of when x== a. 


/ “g—a abi 
ie ’ ® 
ae if x-+2 be substituted for 2 according to the rule ee) » we 
shall have, E " 

Pi ie \ aie. dap ipe? Woda, 

Re" Fae say! wt Bi sel —_ v lL tat) at) 5 ibe fi 

yr sie ‘lak . y - Ki t , NG ke 

* ee 


And, ‘1 extracting the square root of a?--ai, (by the binomial 
theorem,) and then dividing bye 1, we shall Ce | 


e. 


Whence, by making t==0, we find 


> x U = 16, 


¢ 


which is the true value of the expression in the case proposed. 
Ex.5. Leta = (1-2) tan § (va), to find u wheu ¢ = 1. 
In this case u assumes the form 0X o. 


‘i L myer ate uu == ——--+,—,, which bécomes 


oa when « = 1. Matis to this the common rule. 


" a et i 
a a 3 | 
+ hi da, iy 
ak it POs ie) 
dx ~ sin? (we) | 
5 ge Oe ee. 
When « = 1, sin?i{wa) = 15% u = = 
x i , } 
} tan 5 7. — 4 * 
Ex. 6. Let u = ————-—_, to determinate the value iy 
re , wath x? —a?)—* 4 
. , “F eh : + . loo) 
of u when x =a. In this case the fraction becomes =. 
; ab, tal : 
a eh 0 
, 4 


=" 


‘ 
ee 
sal 


ae . a 
Ae ms : 
90 THE DIFFERENTIAL CALCULUS. 
| raga 
Fu 


yuu a ; 
iu i Mh 
But since tanige. — = —-, and 
a x A lt 
Sa cot 17. — wR ‘ 
WW my Gs 
% De. hy 3 
* a a) <1 x? « ey 
va! (22? —a4 ) = ————-< 5 
( PH O(a —a?) I. 
pe 
»_ a(z? —a?)a—* 
oe U ee 3 4 me 
, cot i7. — J ee 
atueneiiete , a . 
Canal ‘eat! ‘ ve 
which becomes 5 when « = a. Differentiating, we find 
rsd ee 
Th ee 
- . d. rn : : * “ 
a dfx — 2a, c—1a 2a (207 me a? exe ioe, 
ln TV aoe bi 
ie. rs ot A 
LEE OS 2 Lie ed 
der es f 
asin? 1 


— 


T 


° 


fi 4a, 
Hence when x = a, u = — 


Ex. 9, Let u = a tan c—}9 sec 2, to find the value of u when 
le In this caseu = w— o. 


But since 
sd 
ii 
tans = » SCC xz = 5 
 cosax 
UE ye T __ «sin x —igr 
ae cota 2cosz Cos x ; 
west > 


. r . 0 
which, when « = 5 becomes 


rr Differentiating, we find . 


d.fx 


: ‘ he 
—— — %.COS & SIn & 
dx & + b) 


d.Fa 


——- = — sine. 
dx 
T ‘ 
Hence, when x = > a Ves. 


Ex" a0) Let ¢ = J fatpan-bat} — J fa nae-te%} 


, a 


hy 


patx. 11. 


Hx.’ 13. 
?. 


Ex. 14 


| er 
» Ex. 15. 


Ex. 16. 


Ex. i 


Ex. 1 


Let = 


» 6 


THE DIFFERENTIAL CALCULUS. 
4 


ww 
w = = 4a, 
, henz =0,u = = fa, 


Let & = = Ya’ =a) ay (a'r) 
» * a—%/(ax*) 


0. 16a 
when x = % (=~ =, 
if) 9 
— enti 
x—2 
Let vu = =: 
—2 
0 yn 
when's =i = = 7 a 
0 r 
De ‘ 
nae —2X ate 


he 
a i 8 
» fey? 

& 


0 
hens =1,uv=>.— = 
w ob si 


se a a 
Va=2) | 


| 0 
when x = Lt 5 = 0, 


Let u=(i— —) tne = =! 
ot x 
h pM oe ee ee 
heh ee 
1 ely a+ 1 
Let x = — = a 
agian Tae (a is 
ON 
whens =1,u=>=5 
1 
OR oi on 
1—x 1—2x? 
sie ain Gila iecad 
© aE TMS Oe 
“(a a? &? — 07)? 
Jifee i= SG? Fa?) (a? 07)? 9 
(gpl: x)(xs a3)? : 
. “y 
wy 


% f i de ' : 
92 THE DIFFERENTISDYCALCULUS. 
i ’ 0 Ee 
when % == a, i _— 
di yl la+al 9 
4 a—-x—alo-alx aa 
Ex. 19. Let « = ——-————,. : 
et a—,./(2ax—x") 
4 al : s 
0 
when « = a,u wh —l, 
LE me ee) 
Ex. Pag Leta = ——__* 
I—atlce | 
a ie : . 1) ny} Ut ey 
Wes when # = 1,u eg tare 


y ey 


SECTION X. 


Application ef the Differential Caliuige is A ihe Theory of shot * 


Curves. 


95. It was in the course of enquiries relative to curve lines that 
Geometers first arrived at the Differential Calculus, which has 
since been exhibited under so many different points of view ; but 
whatever may be the origin we assign to this calculus, it will al- 
ways depend onan. analytical fact antecedent to any hypothesis, 
as the phenomenon of the fall of heavy bodies to the surface of the 
earth is antecedent to all explanations that have been given of it ; 
and this fact 1s precisely that property which all inétiogs possess, 
of admitting a limit in the ratio between their increments and that 
of the variable on which they depend. This limit, whichis diffe- 
rent for different functions, but constantly the same for the same 
function, and which is always independent of the absolute values 
of the increments themselves, characterizes, in a peculiar manner, 
the course of the function in the different stages through which it 
may pass. In fact, the smaller the limits of the independent va- 
viable, the more nearly the successive values of the function ap- 
proximate to each other; the more does the function also approxi- 
mate to coincidence with the law of continuity ; and the more 
nearly does the ratio of its changes to that of the independent va- 
riable approximate to the limit ‘assigned by the calculus. By the 


THE DIPFERENPTIAL CALCULUS. 93 


ba & 


law of continuity is meant that which is observed in the description | 
of lines by motion, and according to which the consecutive points 
of the same line, succeed each other without any interval. The 
method of considering magnitude in analysis: does not appear to 
admit of this Jaw, since we always suppose an interval between 
two consecutive values of the same quantity ;but the smaller this 
interval. is, the more nearly we approach the law of continuity, 
ith’ which the limit accurately agrees : it is also in virtue of this 
law that the increments, although evanescent, still preserve the ra- 
tio to which they have gradually approached, before they vanish. _ 

The preceding statement appears to involve the true and philo- 
sophical explanation of the nature and properties of the Differen- 
tial and Integral Calculus, when viewed in its application to ques- 
tions connected with curve lines and the theory of motion. The 
difficilty, in both cases, arises from the existence of a continuity, 
in the changes of lines and of velocities ; and the -consideration 
of limits, (or any other equivalent to it,) furnishes the means of 
establishing this continuity in thé Calculus.* 

96. Geometrical considerations show very clearly that the ratio 
of the increments of a function and its veriable, is, generally, sus- 
ceptible of limits. 

Every function of one variable may be represented by the or- 
dinate of a curve, whose abscissa is the variable itself; and the 
ratio of the ordinate of a curve to its subtangent, corresponds to 
the differential coefficient of that function, In fact, if in any curve 
whatever, BMM’, (Fig. 4.), we draw through two points M and 
M,asecant M’S; and if we also draw the two ordinates PM, 
PM’, and the right line MQ, parallel to 48; and suppose MT to 
be the paugent tothe curve at the point 7. We have, from the 


supilar amie MQM and MPS, the ratio equal to the 


he 
MQ? 
PM 

vali Ps: : but if we conceive the point MW’ to approach continually 


to the point M, the point S will also approach towards the point T; 


* The above article is taken from Lacroix’s Differential and Integral Calculus, No. 

60. ) ee 

+ We shall adopt the common definition of atangent, which considers it as a line 

‘which meets the curve at the point MZ, but does not cut it when produced on either 
side of this point, at least within assignable limits. Its position at each point must 
depend upon the nature of the curve, and consequenily its subtangent PT, by which 
its position is determined, must be expressible equally with the ordinate y, by some 
function of the other co-ordinate x. 


94 THE DIFFERENTIAL CALCULUS. 


and consequently the line PS will constantly tend to become equal 
to the subtangent PT, and it will just coincide with PT when PP 


: PM 
becomes equal to ngs 5 ; the ratio—- will, therefore, approx- 


PS 
BU . FMS . 
imate to the ratio PT which will be its limit ; and also that of the 
ratio of the increments JM/Q, and M’ Q, which the abscissa and or- 
dinate simultaneously receive. _ Men BS: 


In order to find analitical expressions of those ratios, let AP = 2; 
PM = ¥y, PP =i; andP M = y; then, y = fx, and y= 
f(z+t). We have, as before, from the similar triangles M’MQ, 
MSP, the proportion 


MQ: ine PS; 


heat 5, 4 
iD : ro" 


Now, we have | 

MQ = MP—MP = 7-7; 
but MP’ =y =f (a-+2) 
-, by Taylor’s Series 


d3y 1 


dat Tyo"? ee: 


yayt ge eres 


And consequently, 


, , d2y v 
MQ=yny = Hi Naas" Tat oP 


Substituting this value in that of PS, we shall have a 
PS =~, -, ——— 
dy. , dy 7% : 
a t -- dx?" 1 gt &c 


Ps = ee 
dy dey rt 
ay a ee i 
dz t de i.2.°? oe 


Passing to the limit, 7== 0, and PS becomes equal to PT, we 
have, therefore, 


7? 
hs 
THE DIFFERENTIAL CALCULUS. 95 
dx 
vind PT == =y— = subtangen 
. dy dy 
c * dx bis a 


This is the general formula for determining the ‘subtangent of 
any curve whatever, whether the co-ordinates are perpendicular to 


eachother or not, provided the ordinates are parallel among ae 
ves. 


Let us suppose that the nature of any curve 1M, (Fig. 5,) were 
expressed by an equation containing 2, y and constant quantities. 
If we differentiate this equation, there can never be more than two 
kinds of terms, those multiplied by dx and those by dy. It will 
then be easy, by the common rules of algebra to deduce, from 
this differeptial equation, a value of e which shall contain only 
terms of x, : and constants ; by ‘substituting this value in the for- 


mula at, « ory X = we shall have a value for the subtangent in x, 


y, and constants; finally, putting instead of y its value in terms of 
x, deduced from the equation of the curve, we shall have the value 
of the subtangent expressed in terms of x he constant quantities 
only. 

97. From the preceding equation, we derive 


PM _ dy 


PE a’ 

si | 
It may be therefore observed that the differential coefficient = 

x 


PM 
which expresses the ratio <= PT gives also the value of the trigono- 


“metrical tangent of the angle MTP, which the right line, touching 
- the curve in the point J4, makes with the axis of the abscisszx. 
The co-ordinates being rectangular, the triangle PMT gives 


MT = 7 (TP-+PM), 


ah 9 / io +1 ( 


=_ 


: dx? 
or tangent = f/f } Ya + 


If from the point JV, (Fig. § .) we draw the perpendicular MV 
upon the tapgent MT, the line P.NV is called the subnormal ; 
and MN the normal. The consideration of the similar triangles 
PMT and PMN, gives the subnormal. 


¥ ri 
e 
96 THE DIFFERENTIAL CALCULUS. 
Nin! GPE TR or ely 
DPN = PM. Bar = 9. 
% PT yy og 


“Phe triangle PVN, which has a right angle a at P, gives the ior- 
mal ? 


. d 
MN = J{ PAPE PI = = y/f1- ary : 


98. ‘The following are a fav examples of the applicatio aot 
these formule. ba. ts ia 

Ex. 1. To find the vicbpedand a the fontead parabola, ‘The 
equation of the conical or common parabola being 4? = Lik 38 we 
derive from it, by differentiation, 


from which we deduce 


dy p iflaus ah 
and by putting pz instead of y’, this equation will become 
~ PT or subtangent = 2a. 


Therefore the subtangent 1s just the double of its corresponding ab- 
_ scissa : which is likewise known from other principles. 

Ex. 2. To find the subnormal of the ellipse. The equation of 
the curve, when its origin is taken at the centre of the ellipse, is 


‘This equation, by differentiation, gives : 
Wy 8 Ais) ‘ ! ’ 
putting this value in that of fhe subnormal PV, we obtain 
2 


PN or subnormal = ——32. x 
| a 


Ex. 3. In the curve repreameted by the equation . 


* The equations of the conic sections and the other curve lines which are introdue- 
ed in the course of this work, shall be illustrated in the Appendix, which the student 
ought to read, (unless he is previously acquainted. with the subject,) before he proceeds 
any farther. 


* 
se 
* - 
THE DIFFERENTIAL CALCULUS. 97 
ee | 2 # ’ 
a daxcu-f-y = 0, — 
we have ; ss 
, dy _ dy—2? | jag ew 


and we ey find for the subtangent » 


ydx § yi—axy _ 2axy—2® 


e . Pe ie ay—a? ay — a?’ 


which value may be easily constructed, “when we have assigned the 
value of a, and detained that of hae 
Ex, 4. To find the subtangent of the Cissoid of Diocles, whose 


equation 78. 


3 oe ae 
2=— Hy Pom ae 4d 
J a—x ei % 2 


By differentiation, we have 


2 dy _ 3ax? —2x° 
dx Dylan)? us 
oe Huma)? _ BaF (ame)? 
PY Baa? 22? == Bax? 22° ’ 
and Nhe, 
: subtangent = al zed 
Al hg ‘ an — 22° 


Ex. 5. Let a curve be theh erbola, whose equation, when the 
origin of the abscissa is taken at its centre, is 
: pO oo: “ oe 
ened Colter | 
to find the subtangent. 
oe we have 


SE = youl) _ bial 
; dx azy’ 
Gea ae ee. 
dy b2a “a2 glia 


and consequently, 


Ex. 6. Let the curve be the Conchoid it . icomedes, whose hie 
tron ts * 


(atc) (b°—2°) = ey, 


to find the value of rts subtangent. 


98 THE DIFFERENTIAL CALCULUS. 
By ti we find 
Ly eye (62? —x?)—z (aa)? —2y" | 


i= xy 
an | eg ae vy 
‘Vay dy (a-ba) X(t x (b? — 2 — x?) 2) ma(a-pa)—ay ay? i* 
and consequently, by substitution, aha * 
dp). % x _@ (are)? (6 (U? rail i 


Y dy xoba) X(C—#) 2 (ata (Fay CHa) | 
hence, by reduction, we shall find ; 


x(a-ba)(b? a?) 
ee +b My 


99. Itis often more convenient, and besides more elegant, to 
consider the tangent and normal, by means of their equations.* 
To obtain that of the former, let us examine generally what relations 
ought to exist so that any two lines may touch each other. Con- 
sidering these lines as at first having two points JV and M’, (Fig. 4.) 
in common, it is evident their equations ought to give the same 
values of the ordinate P/M, and the difference M’Q, corresponding 
to the abscissa 1P, and its increment PP’. 

If then x and y denote the co-ordinates of the point V of the pro- 
posed curve, and if we denote by 2’ and y’ those of any point in the 
line which cuts it in Mand MM, we shall have for these two points 


ies 


i Ane dy, a agr 
y = sts Be acts Ke. 


subtangent = 


The second equation is divisible by ¢, and when we take the limit, 
and suppose 7 =0, it is reduced to 


but on this hypothesis, the two points of intersection are united 
into one, which becomes a point of contact for the proposed lines ; 
since they have now only this one point in common. It follows 
from this, that when two lines touch each other, we have, for their 
point of contact € 


* The reader is referred to the Appendix, where the equations of straight lines, 
tangents, normals, &c. are illustrated. 


THE DIFFERENTIAL CALCULUS. 99 


Ma ee 
D> Gee 


When one of these is a right line, whose equation is of the form 
y = Ax'+B, 

and which gives 

te pia 


dx 


we have, for the contact of this line with the proposed curve, 


q 


ta tet _ yy 
y er Ax Sai iz B, A —— de 5 
whence we conclude, that * ‘ge 
ae dy , _ dy ’ dy 
em PS Meer 3 ey ae + y—2 7, 


_ RRC ae 
fo 9=Y Fa. (e 8) ; 


from this equation we deduce that of the normal. which is perpen- 
dicular to the tangent, and which passes through the point .V : this 
will be . 


te 


; dx, 
—Yy =—-—(«i—Zz). 
oY = eee =) 


For instance, in the circle, whose equation is 


‘ y? ++? ea 72, 
we have 
; dy _ C 
dx Yy 


and the equation of its tangent will therefore be 
—y = Fe 2), or yy = mae! sha 


*. yy tax’ = We a= 9, 


The ae of its normal becomes 


SS ae 
mooy (2 a), 


400 THER DIFFERENTIAL CALCULUS. 
which is reduced to 
= ne 


100. When we know how to determine the tangents, normals, 
&c., the following problems may be easily solved. 

Prob. 1. From a given point without a curve, whose abscissa is 
a, und whose ordinate is 8, to draw a tangent to the curve. 

It is evident that we must substitute # for 2, and 6 for y in the 
equation of the tangent, which will @en become 


dy 
Praia (a —<), 


and will serve, when combined with the equation of the curve, to 
determine the co-ordinates x and)y of the point of contact. 

Let us take, for example, the circle, the equation of whose tan- 
gent is 


7 Ae Aba | 
yy pan = 1°, 


we shall have 7 
Byf-axr = 7. 


This equation, combined with that of the circle, will determine the 
co-ordinates x and y of the points of contact ; or, which amounts 
to the same thing, it will determine ‘the points where the circle 
meets the right line, whose equation is 


bytes = 9. 


Prob. Il. From a peint given any where in the plane of a curve, 
whose abscissa is a, and whose ordinate is B, to draw a perpendicu- 
lar to that curve. l 

It is evident that if we substitute « for 2’, and B for y, in the. 
equation of the normal, which will then become 


dx 
bay = — Fle—=), 
\ 3 
and will serve, when combined with the equation of the curve, to 
determine the co-ordinates x and y of the point of contact. 


Let us take, for example, the circle, the equation of whose nor- 
mal is / ; 


/ 


| 


| 
| 


“we shall have . 


THE DIFFERENTIAL CALCULUS. ae | 


nF =: 
8B Pgs or Bx an 


This equation, combined with that of the circle, 


aby? = 
will give 
bk: 3 hu, 
& oo af Ca oe = tar] errr ey 


een ere! Cage) 


the co-ordinates of the points of contact. It appears from the equa- . 
tion 


se 
8% 


i 


f 


y = -2', or Bx = ay, 


8 


that all the normals of the circle pass through its centre, which is 
the origin of the co-ordinates ; and this ought to be the case, since 
the normals to the circle are nothing more Alin its radii. 

Pros. Ill. To draw a line to touch a given curve, and which 
shall also be parallel to a given straight line, or, which shall make, 
with the uxis of the abscisse, an angle whose tangent as represented 
by a. 

It is sufficient to make 

ve th 

dx 
and, by combining this with the equation of the proposed curve, we 
may determine the values of 2 and y at the point of contact. 

Tf the proposed curve be the common parabola, we have 


y= ee os ey amet 
= pt, « Dy a, 
which gives 
Rt BP ise 
3 TDG pne 4a? 


101. In all that procedes, we have supposed the co-ordinates x 
and y to be perpendicular to each other. But it is easy to perceive, 
that when they are inclined at any given angle, the ratio of MQ to 
MQ will still have for its limit that of PM to PT': the equation of 
the tangent will also preserve the same form. With respect to 


102 THE DIFFERENTIAL CALCULUS. 


MT, MR, and PN we may find expressions forthem by means of 
the triangles MPT, MPN, and MT\N, in which we always know an 
angle and two sides, or a side and two angles, 

Definition. ‘Two lines are said to be asymptotes to each other 
when extending indefinitely they continually approach each other, 
and approximate closer than any assignable distance, and yet never 
intersect or touch. 

102. To find a right line which is an asymptote to a curve whose 
equation 1s f (x, y) = 0. 

This problem may be solved by considering the limit of the po- 
sition of a tangent when the point of contact is removed to an in- 
finite distance. 

We observe, that in the curve MX, (Fig. 6.) which has. an 
asymptote RS, whilst the point JM moves further and further from 
the origin of the abscisse, the tangent MT' continually approaches 
to the asymptote; and the points J and D approach the points R 
and £; so that AR and AE are the limits which the values of AT 
and 4D cannot exceed, nor even attain; but to which they may 
approximate as near as we choose. From this it follows, that in 
order to discover whether a curve has any asymptotes, we must 
find whether the expressions for .4T and 4D, relative to this curve, 
are susceptible of limits : if that be the case, and these limits be 
determined, we shall know the two points R and E through which 
the line RS must be drawn, which will be the asymptote required. 

The expressions for 17’ and 14D may be deduced from that of 
PT; the first by observing that AT =. P—PT; the second by 
means of the similar triangles .1.DT’ and MPT: they may be also 
deduced from the equation, 


BO ees dy 1 ah 

y dy A ayy 
of the tangent, by making successively y = 0 and 2’ = 0; we 
shall find 


If, when z is increased without limit, these quantities have limits, 
the curve has asymptotes, and they will be determined by these li- 
miting values of AT and AD. If AT havea limit, but 4D none, 
the asymptote is parallel to the axis on which this latter is measur- 


ed. 
In order, therefore, to determine all the asymptotés which any 


proposed curve ought to have, we must successively make x and y 


THE DIFFERENTIAL CALCULUS. 103 


infinite and afterwards substitute, in the expressions for AT, and 
AD, each of the different results which these two hypotheses af- 
ford. When AT and 4D are always infinite at the same eng we 
may conceive that the curve has no asymptote. 

If the limits be impossible, the curve has no asymptotes. 

If AT = 0, the axis of y is an asymptote ; and if the limit of 
AD = 0, the axis of x isan asymptote. If both limits = 0, the 
asymptotes pass through the origin, and their direction may be 


d 
found, by the limiting value of os which represents the tangent of 


the angle MTP for any point of the curve, and we shall thus get 
the tangent of the angle SRB. 

103. If we apply these principles to the general equation’ of 
lines of the second order 


# 


y? == me-bn2”, 


we shall find 


2y? ma 
m+-2nx n+2na ” 
mx-+ 2nx2 mx 
AD =y — - == 


By mana)’ 
The second, member of these equations being put under the form 


# 


their respective values, when x becomes infinite, are seen to be 


gam AR and gy = = AE. 
if » = 0, the expressions for 4T and AD become infinite at the 
same time with x ; and the curve has no asymptotes ; nor will it 
have any, when a is negative ; because, in that case, its equation 
will not admit of an infinite value of x ; for the value of AEH be- 
comes in that case imaginary. Now, in the case in which n is ne- 
gative, the equation belongs to an ellipse ; and consequently this 
curve has no asymptote. But this same equation becomes that of 
a parabola, when n = 0, in which case we have 
mm 


Mm ' 
AT a = 7, AD = = = M5 . ie 


104. THE DIFFERENTIAL CALCULUS. | 


which proves that the parabola cannot have asymptotes. 
104. In the curve represented by the equation 


Crom Sary - y> = 0, 
we have m 


AT = axLy WATS axy 


4 
parent 


x? —ay? y? —ax 


in order to find the limit to which these expressions approach 
whilst y increases, we must substitute for y the limit to which it, 
tends, and must consequently know the value of y in terms of « ; 
we may, however, in the present instance, supersede the necessity 
of obtaining this value, by avery simple artifice. If we make 
x = ty, the proposed equation becomes divisible by y? ; and we 


have 
Sat 


| Te 
itis readily seen, that the supposition of ¢ =—1, will make y in- 
finite, and will give = —y. Changing z into —y im the expres- 
sions for AT and AD, and then taking the limits, we shall have 


AR =—-a=JAE; 


and drawing through the points Rand E, (Fig. '7), determined by 
means of the preceding values, the line RE, it will be the asymp- 
tote of the branches AY and 4Z. 

105. To determtne the differential of the arc of a curve, consi- 
dered as a function of the co-ordinates of its extremities. 

By the equation of the curve, yis a function of x: let 4P=a, 
(Fig. 4.) PM = y, PP =1i,PM =y =f (e+72); .. by Taylors 


- series, 


d2y 42 
PM — Pie a oy aa Me al 


ie dz dz? 1.2 
The co-ordinates being rectangular, MM = y (22 ay M'Q?) ; 
.. the value of MM’ must have the form ity 
| MM = $2 (ptPIrit%, 
by making ‘ # 
oa dy i. dy 42 d3y q° uh : 
ioe and 2 e aa Te Xe. = P22. 


The limit of the ratio of the are MOM’ = s, and its chord being 
a ratio of equality (35), it is evident, that when? = 0, 


yf 


: ‘ ° 
SHE DIFFERENTIAL CALCULUS. 105 
MM’ ds 
5 dx 
But 
‘ MM 


=f + @ 4+ Pit} 


@ 


consequently, when? = 0, 


MM' 
2 Si Lge 
ee de v}1 raat 


Hence, ds = V §du?+- dy? 


the circle whose equation is 


et-- y= a, 
giving 
2 dt + y dy = 0, or dy = — oe there results 
2 dnp? 
i= V jae ti} aS yety)s 


therefore, by putting a for «*-+-y’, and ,/(a°—x’) instead of y, we 
have 


adx 
Wes | 
which result is conformable to that of art. 43, when we suppose a 
or radius equal to unity. 

106. To express the differential of the area included by a curve, 
aud the ordinates of any two points upon it, asa function of the 
co-ordinates. 

Since the are MOM’ (Fig. 4,) and chord MM’ coincide when 
i or MM is indefinitely diminished, the limit of the ratio of the 
area included by the ordinates and the arc MOM’, to that in- 
cluded by the or-dinates and the chord MM’, is a ratio of equality. 
Let da be the differential of the area; it is evident that when ¢ = 0, 

PMMP _ da 


——— oa 


4 de 


tie 


But since the area 


106 THE DIFFERENTIAL CALCULUS: 


PMMP’ = PP’ Xx\(PM4+P'M) =hilyt+y), 


dyi ,d?y # 
and y Sat sk eit de 12 Bre. 
i. dy 2 tg 
“2G ty) = YT oe woes ae ‘i 


Hence the area ‘of P.MM'P’ is expressed 7 


dy 
il y (sty) yt of dx2 ms 55 aap Xe. 


PMMP _ dyi ,d?y # 
oc tyte Nt dee dea ge 
When i = 0, we shall have * 
sa 6 da = ydx 


In the circle 
da = dz (a*—x?), 


thus, though we cannot assign the algebraic expression of a circu- 


lar segment, yet we may arrive at that of its differential, from the — 


me. 


consideration of limits. 
107. We shall conclude this section with a few’ practical exam- 
ples. 
Ex. 1. Let the equation of the curve be aa2-+ay-+e—y’ = 


; 


3yP—2a1 
subtangent = pula 7 


2ar+y? y2-- 3x? 
Kix. 2. Let the equation of the curve be 
__ ay (ax —2”*) 
errr : 


2(ax — x") 
subtangent == — -———-—-; 
a 


\ ‘gia 


ae to the tangent 


SS 


* Bee ( bags 


Equation to the normal 


ie. oe) § a! ie 


a 244 


en Ee Sax! +-2a7—3ax§; 


-, 


THE DIFFERENTIAL CALCULUS. 107 


A line passing through the origin of the abscisse, at right angles 
to the axis, isan asymptote to the curve. 
Ex. 3. Let the equation of the curve be 


ye — ax?-ea?, 


A line whose equation is 
, Ul a 
y =a--e, 
| 3° 
is an asymptote to the curve: this may be constructed as 


follows; make 4R = AE = =i then, the line passing through R 


and E is the asymptote required. 
Ex. 4. Let the curve be the hyperbola referred to its axis, whose 
equation is 


63 . 
y? = — (2ax -+ 2°). 
ah 
The linear equation 
by 
ee “fe te ; 
y Hi ps ue be at, 


determines the position of the two asymptotes of the curve. 
Ex. 5. Let the curve be the cissoid of Diocles, whose equation 
is 
3 


= 4 (8a — 22) 2 "ax. 
2(a—x)? 
If « = a,y' becomes infinite and coincides with the tangent, or, 


in other words, is an asymptote to the curve. This may likewise 
be shown by the method given in art. (103). 


108 THE DIFFERENTIAL GALCULUS. 


SECTION XI. 


The determination of Singular or Remarkable points wf Curve 
ibs Lines. 


108. Those points of a curve which are distinguished by some 
remarkable circumstance, are sometimes called singular points. 
Before we proceed to show the methods of discovering their ex- 
istence and of determining their position, it is necessary to explain 
the geometrical signification of differential coefficients. It has 


been already observed, art. (97,) that a gives the value of the 


trigonometrical tangent, of the angle, which the right line, touching 
the curve in any point M whose co-ordinates are x and y, makes 
with the axis of the abscisse. This proposition may be demon- 
strated, a prior?, in the following manner : let PM = y, (Fig. 4,) 


PP’ =7; by drawing MQ parallel to the axis of the abscisse, we 
have therefore, - 


MP =f (x + 1%); - 
Le : a , d2 ;2 
’ MQ=f(e+ij-fe= Foot ate re + We. 


Now, making radius = unity, 


MO: MOQ-*1: ee 
Ard th Ma ys 


and by putting for the expressions MQ, JMQ, their values, we shall 
have 


“4 a 1. ot ke 


Passing to the limit, 7 = 0, and tangent S is changed into tan T'; 


dy ; 
it tH Se aS. 
.,. tan rE 


Yo ee cee F 
This being premised, ifthe differential co-efficient = which 


- expresses the tangent of the angle MTP, becomes evanescent, it 


oe 


THE DIFFERENTIAL CALCULUS. 109 


follows, that the right line which touches the curve at the point 
M, is parallel to the line of the abscisse ; and also, that if it 
change its sign after this point, the tangent is then inclined towards 
a different side of the ordinate to what it was in the former case. 

An examination of the two figures 8 and 9, will show that, in 
this case, the ordinate, after having attained a certain magnitude, 
begins to diminish (Fig. 8) ; or otherwise, that after having dimi- 
nished to a certain point, it begins to increase. (Fig. 9.) 

The first circumstance corresponds to a maximum value of the 
ordinate, and the second to a minimum value. When either of 


d 
these takes place, we have equally “2 = 0, as we have also shown 
> as 


from analytical considerations, since any equation, y = fx, between 
two variables, may be always considered as the equation of a 
curve in which the different values of the function y shall be the or- 
dinates. 


J is positive 


2 
109. Let us now examine in what circumstance, aoe 


or negative. 

For this purpose, let us consider, in the first place, the case m 
which the curve (Fig. 10,) is convex towards the axis of the ab- 
scisse. 


Let AP =a, PM = y, PP’ =7; and through the points M and 
M' draw the secant MM'S, and the straight lines MV, M’N’, paral- 
lel to the axis of the abscisse ; we shall have 


MO = M'P'—MP = f(x-+i)—fr ; 


or, which amounts to the same thing, 


MO: MN :: MO: SN, 
or 
a: June: MO: SN; 


. oN = 2 MO; 


and, by substituting for JM’O its value, we have 


eB: gay. 8 
SN = Bt + ea 1 &e 


Li0 THE DIFFERENTIAL CALCULUS. 


Again, since . 
te 20 d2y 4i2 


MUP" = f (a+2t) = y asi dn? 5 +, Xe. 
and VP" = PM = Y 5 

4? 

“i MN = es = 2 hs <= +, &e. 

br ada? 

and eiaatonte 
2 
M'N-SN = M'S = a | 2 +,&ce... 0. (1). 


But if the curve is concave towards the axis of the abscisse, 
(Fig. 11,) we must subtract the value of M’N from that of SN, 
we have therefore in this case, 


12 
M'S=——it,&e. . . . . . + (Q). 


By comparing these two values (1) and (2) of MS, we see that, in 


d21 
the one, — cee. tis affected with the sign -+, and in the other, with 


the sign—; and as 7” is essentially positive, and the first term of the 
development may be rendered greater than the sum of all those 
which succeed it, we may conclude that, when the ordinate is po- 


Pl! ak is : i > 
sitive, 4 is negative or positive, according as the curve is con- 
cave or convex to the axis of the abscissx 
The inspection of the curves cm, (Fig. 10), and (Fig. 11), 


2 
placed below the axis of the abscissz, shows that the signs of ka 
ought to be taken in an inverse order, when the ordinate is ne- 
gative ; and that consequently, a curve is convex or concave to- 
wards the axis of the abscissa, according as the ordinate and its 
second differential coefficient are of the same or different signs. 

_ 109. The curves which have been just discussed, are supposed 
io be situated either aboveor below the axis of the abscissx ; but in 
case that the curve, which was at first turned towards the axis of the 
absciss, is afterwards turned the opposite way ; this circumstance _ 
is what is called an inflexion, and the point JM (Fig. 12,) is called 
a point of inflexion or of contrary flecure. It may be recognised 


Sues : ee 124 
by the change of sign which the coefficient a undergoes, before 
rv 


and after the point WV; it may also be determined, by seeking the 


THE DIFFERENTIAL CALCULUS. iil 
position of the curve with respect to its tangent before and after 


this point. 
-\ The equation of the eS being in general 


f, =e dy, t 
y—y) = da ee): 


we shall have, making 2’ =2x-+7, 


i 
dy. 
Mie aralceeghs” 
dy. 
or y= yt - a, 


for the expression of P’V’ (Fig. 13.) which is the ordinate of the 
tangent, corresponding to the point P’, whose abscissa is x2 ; but 
since y is a function of x, we have, for P’M,, this series 


e dey OP 
yt Bite T5 +—4 0.3. -+, &c. 
| whence we deduce 
d2y BR diy 


os i 3 

PM— ith a sca ils Seuiy doads Zbl’ oki 

Again, taking a point P, upon the axis of the abscisse,' behind the 
point P, and whose abscisse is x—2, we should, in like manner, 


find 
eg ea ay 
main, eile as wiRPA eet DIGI a 


sp Vey ORGY 
Now it is evident, that if w = 0, the two differences 


PP os sf Pipl hat: Ba 
diy 0 
P a pet ed 
M—PN, =—54 so 4, &e 


will have contrary signs, at least when: is taken so small that the 
first term of the series shall be greater than the sum of all the 
rest ; and thus the proposed curve, after having been situated in the 
Biter part of its course; below the tangent, will now pass above 
it, and wice versa, 


112 THE DIFFERENTIAL CALCULUS. 


There will, therefore, take place an inflexion at JM, and nota 


2 


d2 


d Ps ‘ q . : 
maximum or a minimum, if “5 wppish at that point, together 


with a and in general, if the first of, ta differential coefficients 


which does not vanish, is of an odd order. Such is the geometrical 
ing of the analytical characters indicated in art. (77). 

110. An inflexion may take place at a point where the tangent 
is not parallel to the line of the abscisse ; and where, consequently, 


d f . : : Hibs 
. does not “gs but what always characterises this point is the 


change of sign aro Pe ! with respect toy. 


It is evident, ae ae y integral quantity can only change. its sign, 
by passing through zero; but a fraction may also change its sign, 


es ie hs eae 
in its passage through infinity, as happens, for instance, to which 


successively becomes 


O@ yoo 

b 4 0’ ane EF 
when xis made = b,x = 0, c= —b: we may, therefore, con- 
clude, by what has bean said, that at a point of gis Aexure, 


d2 
at is nothing or infinite ; but we cannot reverse this proposition. . 


f dy”. re tet 
111. When a instead of vanishing, becomes infinite, the or- 


dinate becomes a tangent, as at E (Fig. 14,): this circumstance 
indicates a limit of the curve, in the direction of the ‘abscisse ; 
that is to say. a maximum or a minimum of the abscissa provided 
no inflexion of the curve take place at that point where the tangent 
is perpendicular to the line of the abscissz. 

For example, let us take the equation 


yy =azr—, 
by differentiation, we derive from it 
dy  @ 
dc 2y , 


This value, being put equal to zero, gives y = wm ; therefore the 
curve cannot have a maximum inthe direction of the ordinate. 


| THE DIFFERENTIAL CALCULUS. 113 


a ti gl | 
but if we suppose infinite the value of a we would have o = o, 


a condition which is fulfilled | by making y= 0; ; in which hypo- 
2 
thesis, the value of Si is reduced to =a positive result. We see, 


therefore, that the value y = 0 corresponds, to a minimum of x. 
We can determine this minimum by making y = 0 in the proposed 
equation, which will reduce it to 


ax—b = 0; 


whence we derive 


Rl oe 


for the minimum sought. 
d ages 
112. We have seen (91), that 2 becomes infinite, when some 


radical quantity disappears from the expression of the function y. 
it must be noticed that at that moment the quantity by which this 
function changes by the variation of x, must, contrary to the ordi- , 
nary rule, have more than one value corresponding toa single value 
of y; for, unless this were the case, we shouldnot again get the num- 
ber which the degree of the function entitles us to, and which must 
alwage remain the same, the equality of several of these values 
being, necessarily, only momentary. 

This circumstance takes place at the point E (Fig. 14), when 
itis evident that, for the abscissa 1c, consecutive to C, the curve’ 
has two ordinates, and consequently the same ardinate CE has two 
differences, the one ce’-—CE, and the other CE —ce. 

The same thing equally happens at a point, as G, where the two 
branches of the curve cut each other; that particular ordinate FG 
has also, for one and the same increment Ff of the abscissa, two 
differences ; the one fg’—FG, and the other FG —fg ; but in other 
points of the curve, one and the same ordinate has, for each incre- 
ment of the abscissa, but one single difference. 

_ The points where several branches intersect, as G, or the junc- 

tion of two branches, GDE and GD'E, as E, are called multiple 
points. They are recognized by one ordinate having, for the same. 
abscissa, more than one differential, which causes the differeritial 
coefficient to become infinife at.those points. Itmay also ex- 


114 THE DIFFERENTIAL CALCULUS: 


hibit itself under the form x which always happens when its ex- 


pression contains, at the same time, both the variables x and y. 

At each of these points the curve has more than one tangent. 
At G, for example, it has two distinct ones; at E ithas also two, 
but united in one, which is the limit of those of the superior branch 
‘GD'E, and the inferior GDE. 

113. The ‘multiple points sometimes assume two particular forms, 
to which have been given the names cusps, or points of reflexion, 
because the branches of the curve which meet them, extend no 
farther, and the curve is, as it were, bent backwards. That of 
(Fig. 15.) where the convexities of the branches are opposed, is a 
cusp of the first species, and that of (Fig. 16.) where their conca- 
vities are turned the same way, is a cusp of the second species. 

These points have only one tangent, but which must be looked 
upon as double, like that of the point E in the figure of the pre- 
ceding article; and they are distinguished from other multiple 
points, by the course of the curve before and after them, with res- 
pect to its tangent, and which may be recognized by the sign of the 

2 
differential coefficient se (109), when we take successively for a, 
values greater and less than the abscissa corresponding tothe mul- 
tiple point under examination. 

114. All which have been said may be reduced toa rule as sim- 
ple in its enunciation, as it is unequivocal in its application: tke 
determination of the abscissa, to which a singular point corresponds, 
ts obtained by inquiring, when the differential coefficients, of what- 


t ape A Oy) F : 
ever order, become nothing or infinite, or 5: the species of the point 


is assigned Ast, by examining what number of branches passes through 
this point, and whether or no they are extended on both sides Of it ; 
Qdly, by determining the position of their tangent ; and 3dly, the 
directions in which their concavities or convexities are turned. 

115. In the family of curves, ras by the very simple 


equation | 
‘igu bbe (za), , 


we shall find examples ‘of almost all the particulars above enume- 
rated ; and their discussion is very proper to illustrate the rule there 
delivered. The expression 

dny j 

zo me (mm)... . (m—n--1) ¢ (a—a)™—, 
vanishing by the supposition of « = a, in every case where m>n, 


THE DIFFERENTIAL CALCULUS. EIS 


and becoming infinite when m<n, it follows, that the abscissa a 
corresponds to a singular point. 
The exponent m may be positive or negative, greater or less, 


than unity. We shall at first suppose it positive and > 1. 
If it be an éven number, or a fraction with an even numerator, 


we find, Ist, the same value of y, whether we take <a, or >a ; 
the curve, therefore, pursues its course aboye the absciss@, which 
precede and follow a; and there passes only one branch through 
the point under oxalninntigie 2dly, from the expression 


dy = me («—a) mai 

dx 

which ee Che ean x = da, we see that the tangent at this point 
is parallel to the line of the abscissz. 

3dly. If we make « alternately <a and >a, in the value of 

i =m (m— 1) C (ram a)\"—?, 

and if we notice that, on the stitypeltitioni we set out with, the ex- 
ponent m—2 is also an even number, or a fraction with an even 
numerator, we shall perceive that this differential coefficient retains 
the signin both cases with the ordinate y itself; and of course 
the curve turns its concavity the same way, on both sides of the 
point we are considering ; its course, beyond this point, is, therefore, 
of one of the two kinds represented in (Fig.17) ; the first if c be 
negative, the second, in the other case. 

If m be an odd number, or a fraction whose numerator and de- 
nominator are both odd, the ordinate corresponding to each ab- 
scissa, has only one value ; and by taking «<a, and x >a, we find 
for y two real values: the curve is, therefore, continued on both 
sides of the point we are examining, and it has but one branch 
passing through this point. The tangent is, as before, parallel to 
the line of the abscisse ; but the exponent m—2, being now an 
odd number, or a fraction with Bp odd numerator and denominator, 


9 


d? 
the coefficient = will hence: its sign when x is made suc- 


cessively <a and >a. The curve in consequence has not its 
concavity turned the same way on both sides of the point under 
consideration: this point is, therefore, one of contrary flexure, as 
in (Fig. 18.) | 

Lastly, if the exponent m be a fractional number, whose deno- 


BR 


116 THE DIFFERENTIAL CALCULUS. 

minator is even, the quantity (a—a)™ being susceptible of the signs 

-L, the ordinate y will have, for every abscissa, two’ real values, 

when zis greater than a, and only imaginary ones whenv<a ;. 
two branches, therefore, of the curve passes through the point un- 

der consideration ; but which extend only on one side of it. The 

tangent J still parallel to the axis of the abscisse. ‘The coeffi- 


cient ie Y has two values with opposite signs, while those of the 


ordinate have the same. . 
Hence it follows, that one branch turns its concavity towards 
the axis of the abscissw, and the other its convexity, as is shown 


in (Fig. 19.), which produces a cusp of the first species. 3dly. 


If the exponent m< 1, since we should then have — 


the value 2 = a will render this differential coefficient infinite ; 
and the line which touches the curve at the point where x = a, will 
be perpendicular to the axis of the abscissee. We shall find also 
by considerations similar to the foregoing, that the point C is a li- 
mit of the curve in the direction of the axis of the abscisse, when 
mis a fraction, whose numerator is odd, and denominator even : 
that this point is a cusp when the numerator is even, and a point of 
contrary flexure, when the numerator and me Ngee are. both 
odd. 


The ordinate y would become infinite, ahd be changed into an 


asymptote, if m were negative. 


116. The curve represented by the equation 
(yrat) = ot 


offers an example of a cusp of the second mo es ‘An this 
curve 


5 
y = eck? 


in order to know whether it has any singular point, we must 
enquire whether any of the differential coefficients of the function 
y become nothing or infinite. We first obtain 


dy 3 ths 5 3 6 


THE DIFFERENTIAL CALCULUS. riz 


the first of these results vanishes when x = 0 ; the second reduces 
itself to 2, and we see also that the third differential coefficient 


d3y 5 3 1 

——= = tis ,-. — 

ds | -yeng ee” 
becomes infinite in that case; the point corresponding to 2 = 0 
is, therefore, a singular point. Itis evident that this point is a limit 
of the curve, which does not extend on the side of the negative ab- 


: i eh ae 
scisse, since the term x? then becomes imaginary. The values 
, ° d2 e,° ® 
of the coefficient are both positive when x is very small, and 
& 


are of the same sign as those of y; both branches of the curve, 
therefore, turn their convexities towards the axis of the abscissx 
AB (Fig. 16); they touch at 4, for they have 4B for a common 


tangent, “ vanishing at this point. It results from all these cha- 


racters taken together, that the form of the curve at this point is 
such as the figure represents. 

The preceding examples relating only to those singular points, 
where the branches of the curve touch one another, afford instances 


of a single tangent only. ‘The curve corresponding to the equa- 
tion 


ay = 4/ (ates 2 x} 


presents, at the point where « = 0, two brdlighes which cut one 
another: but we shall not dwell on this example, because we shall 
discuss, ina subsequent part of this Section, another in which the 
same circumstance takes place. 

117. Curves are sometimes accompanied by insulated points, 
which have the character of multiple points; but may be distin- 
guished from these by this, that in the case of the former, the co- 

ae mt lowilee | 
efficient = assumes an imaginary value. 

Take the equation , 

ay’—a3 --bx? = 0; 


from which may be obtained — 


i a oar 


ao 


- 


_ 


118 THE DIFFERENTIAL CALCULUS. 


dy __ x (3n—2b) 
dx 2 /$ax(x—6)? oe 


The differential coefficient, when “% == 0, becomes _ but its true 


value may be had by suppressing the factor x, common to the nu- 
merator and denominator : thus we obtain 


dy 4 32—2b } 
dx 2/$a(a—b)} K 


whence, making + = 0, there results 


dy. 2b 
dtvodderakie 


an imaginary expression. 

Upon the same supposition the proposed equation gives y =0 ; 
but this ordinate, which is imaginary when 2 is negative, becomes 
so again, untila =6. Thus the point 1, (Fig. 20) although com- 
prised in the equation, is absolutely detached from the curve. 
Points of this kind are called conjugate points: they result from 
certain finite portions of the curve vanishing, owing to the parti- 
cular value of some constant in the equation. 

The curve represented by the equation 


ay? —x°-+(b—c)x?-+-bex = 0, 


which gives — 


yet fee y 


offers an example of these changes. Its course is at first, as re- 
presented in (Fig. 21); the supposition c = 0 reduces the part 4F 
to the single point 4 (Fig. 20), as we have seen above. When 
b = 0, (c not vanishing), it assumes the (Fig. 22,) andif b = 0, 
at the same time that ¢ = 0, the (Fig. 23). 

Curves have also occasionally singular points, which are not, vi- 
sible: they are such as result from an even number of inflexions, 
uniting into one. See, for these points, and for those of serpentement 
or of undulation, from which they take their origin, the Traite du 


Calcul Differential et du Calcul Integral, 4to. ; or Peacock’s Ex- 


amples of the Applications of the Differential and Integral calcu- 
lus. A few examples of those points will be found towards the 


THE DIFFERENTIAL CALCULUS. 119 


conclusion of this Section. We shall now proceed to give an ex- 
ample of the analysis of a curve. ; 

118. We divide lines into, different orders, according to the 
degree of their equations. The right line constitutes the first order, 
since it is represented by the general equation of the first degree, 
involving two indeterminate quantities. The lines of the second, 
and of the third order, are those whose equations are of the second 
or third degree ; and so on for others.. Newton, considering that 
the first order included only the right line, and that curves did not 
exhibit themselves before the second, divided these latter into 
classes, and called lines of the second order, curves of the first 
class; those of the third order curves of the second. class, and so 
on for the higher orders. Lines of the same order are | subdivided 
into species, from a consideration of the principal circumstances 
which characterise their course. 

Were it possible to resolve equations of every degree, there 
would be no difficulty in tracing the course of a curve represented 
by any algebraic equation whatever. In fact, if we suppose, that 
this equation being resolved with respect to one of the indeter- 
minate quantities, y, for instance, which it involves, should furnish 
the different roots X', X”, X"", &c. which will be necessarily 
functions of x, and constant quantities ; the question will then be 
reduced to the particular examination of the courses of the lines 
produced by the equations 


y=XMyy =X, y= X"", &e. 


when we give to « every possible value, both positive and negative, 
which the functions X’, X", X’”, &c. will admit of, without be- 
coming imaginary. These lines yea} constitute so many branches 
of the curve which is represented BY the proposed equation. 

119. The extent of each branch will be determined by the ex- 
tent of the limits, between which are comprised the different so- 


' Jutions of which the particular equation by which he fi represented, is 


susceptible. If amongst the quantities X’, X”’, &c. there be 
found any which become infinite, or in el we OR z to be 
infinite, there will correspond to them branches, whose course 
will be infinite, since they will recede to an infinite distance from 
one, and sometimes also from both of the axes to which the curve is 
referred. 

A branch of a curve never terminates, unless the expression for 
its ordinate become imaginary, though it does not therefore fol- 
low, that the course of the curve is not interrupted ; it only hap- 
pening, that in this case, two branches are united, and are recipro- 


{206 THE DIFFERENTIAL CALCULUS. 


cally continuations of eaeh other. We may be easily convinced 
of this fact, by observing that the number of imaginary values of y 
is necessarily even, and that each pair of them consisted of real 
and equal roots, before they become imaginary. In fact, the pro- 
posed equation being always decomposable into factors of the first 
‘and second degree, if we represent one of these latter by 7°— 
2Py+ Q = 0, we shall find that its roots P + 4/ (P?—Q) are not 
imaginary, unless Q be greater than P?, than which it was origi- 
nally less; and that there must be a point where the functions of 
a, which are designated by the letters P and Q, are such as to give 
Q = P23, which will annihilate the radical quantity, and give to ¥ 
two equal values. 

120. Let us take the equation 


y'—96 a242-+100 a%a2—x! = 0. 


This equation, which is resolvible, both with respect to « and 
to y, gives, in the first case 


y = af 48a bo (23040 10002a? +2")}. 


; - e e / 
To determine the limits of the curve in the direction of the or- 
dinates, or to discover whether y is susceptible of a maximum or a 
minimum, we must examine in what case the differential coefficient 


becomes equal to nothing ; we shall then have 
a—50ax? = 0; 
whence 
a = 0,2 = + 5a,/2. 
The first value of x substituted in the proposed equation, gives 
y= Oandy = + 4a,/6. | 


The two values of y, which are equal to + 4a,/6, determines 
the points D and D’, (Fig. 24) ; the one situated above, and the 
other below, the axis of the abscisse, and which are also maximum 
values. We may easily convince ourselves of this, by finding 


d? . ; ; j 
the value of “i corresponding to this hypothesis, or by showing 


from the expression for y, that the values of the ordinates which 
immediately precede and follow it, are both less than 4a,/6. 


THE DIFFERENTIAL CALCULUS. 121 

121. The concurrence of the two values 7 = 0, atid 4 = 0; itt 
dicates the point 4,and makes, at the same time, on =F To dis- 
cover the value and import of this last expression, which i in general 
characterises a multiple point, we must have recourse to the process 
in art. (92%); but this may also be effected by finding the differential 


coefficient of the second order. For this purpose, we observe, 
that the first differential of the proposed equation is 


(y—48a7y) dy + (500% —a°) dx = 0; 
and the second differential 3 
(y°—48a°y) Py--(3y?— 48a?) dy?-}-(50a?~ 32") da? = 0; 


and that, in the case in which « and y are evanescent, this reduces 
itself to ; 


— 48a7dy? +- 50a? dz? = 0, 
which consequently gives, for this case alone, the values of the 


ee ; ae 
coefficient —, which we were not able to deduce from the first dif- 


dx 
ferential: we thus get 
a 50 45 JS? 
Res Pee MMR e') Ba a 


It follows, from these values, that! the curve has, at the point 4, 
two tangents, which make, with the axis of the abscisse, angles, 
whose trigonometrical tangents are respectively 


5/2 
> Fan ov 3? 
and which consequently admit of a very easy construction.” 
122. There yet remain, to be examined, the two roots 
a= + 5a,/2. 
By substituting them in the proposed equation, they make y 


dy . 
* We shall succeed in general, as above, in finding the true value of oa in the case 


0 
in which it becomes 5; by examining the successive differentials of the proposed 


equation, and by age up to that order whose exponent is equal to the maher 


wed vs 


of values which a = Ought to have. 
R 


122 THE DIFFERENTIAL CALCULUS. 


imaginary, and consequently give neither a maximum nor a wu- 


mmum. 
123. To obtain the limits of the curve in the direction of the 


abscissz ; or, what amounts to the same thing, to find the maximum 
and minimum of x, we must make the denominator of the fraction 


which expresses o equal to nothing, which will furnish the equa- 


tion y°— 48 ary = = 0, mange: we. and y = & (4822). The 
first value gives 100 a°z?— x* == 0, from which we deduce xz = 0, 
and « = = 10a. 

The root x = 0, again indicates the rltiple point placed at the 
origin 4; but the other two roots correspond to the points J and 
I', where the curve meets the axis .4B of the abscisse, and which 
hace not yet been remarked. 

The last two values, y = + 4/(48a*) = + 40,/3, fend us to 
x = -b 6a, and x = + 8a; the one of these results enables us to 
recognise the point F, and those corresponding to it in the other 
branches ; the other determines the point H with those correspond- 
ing to it likewise. We may also observe, that at the points F and I 
the abscissa is a maximum, and at the point H, a minimum ; since 
the curve turns its concavity in the first case, and its convexity in 
the second, towards the axis AC of the ordinates. 

124. To complete the determination of the principal circum- 
stances which distinguish the proposed curve, it yet remains to 
inquire into the nature of its principal branches, and its different 
points of inflexion ; for knowing its multiple points, we know al- 
ready that it has no cusp or point of reflexion. 

We shall begin with discussing the nature and number of its in- 
finite branches. We may easily assure ourselves, that the 
values of y, in (120), become infinite at the same time with 
«; but without recurring to these values, if we make y = ta 
the proposed equation will be divisible by «, and will thus become, 


3 


t4 0? —9607t?-+-100a?—a? = 0; 
whence we deduce 
_. 100a*— 96070? 
1 thes 


a result which gives « = + an infinite quantity, when? = 1; in 
which case also y = a 
We shall also have (102) 


T dx il a’ —~ 50072? — y*-+-4 Gay" 
4 dy 9 wm 50070 


THE DIFFERENTIAL CALCULUS. 123 


fs dy yj — 48074? — 2*+-5007x? 


Yum OF NES Se SN 


dx pa 48a*y 


These expressions, when we substitute the value of 2*, become 


500727 —4807y2 4807y? — 50072? 
w—-500%e § y—48a7y 


? 


which diminish continually, whilst « and y increase, and are ace 
tually evanescent, when we suppose y = x. We thus see (102), that 
the asymptotes of the proposed.curve are two right lines drawn . 


d 
through the origin 4; and as the expression for “- has unity for 


its limit, it follows that they must make an angle of 45° with the 
axis of the abscisse. We have not drawn them lest the figure 
should be too complicated. 

25. We now proceed to find the inflexions or points of contra- 
ry flexure. Wehave > 


dty ( @ ) ( y eo 


dz? yp—48a7y 


: 0 | 
this expression becomes ry when «x and y are evanescent, a case in 


., dy? 50 Mel. malta 
which a = mk and to determine its true value it will be ne- 


cessary to find the third differential of the proposed equation. 
Making, in the result, 2 and y equal to zero, we have simply 
— 144a?dyd’y = 0, 
which gives 
d2y 


daz? 


and proves that the point .4 is, in fact, a point of contrary flexure. 
To discover whether the proposed curve has any others, we 
d?y 


must make the numerator of the expression daz C1ual to zero, and 


there will result the equation. 


dy? 
327 — 50 07 ae (3y° = 480") TS = 0 5 


hg a ae . 
putting for ao its value, and making the denominator disappear, 
we shall have 


124: THE DIFFERENTIAL CALCULUS: 


(32° — 50a?) (y?—48a°y)? 
— (3y?— 48a") (2° — 50a%x)? = 0: 
we may give to this equation the following form : 
4P(y? ~48a")?(3y? — 500") 
— 2727 — 60a —48a") = 0. 


% 


If we afterwards obserye that the proposed equation is reducible to 
the form " 


(y?—480°)? —(2?—500t)°+-1960! = 0, 


and if we deduce from this the value of (y’—4%a*)*, for the pur- 
pose of substituting it in the equation preceding, we e apt! and, af. 
ter proper reductions, ' ; eo ie 


(2?—600?)?(25y?— 2427) +- 984? y°( 3a? = 50a") = 


this last equation, combined with the proposed one, will serve to 
determine the abscisse and ordinates of the point of contrary flex- 
ure K, and those which correspond to it in the other branches ; 

we shall be easily able to deduce from it the value of y*; and by oe 
substituting for it in the equation of the proposed curve, we shall. . 
have a result which involves x only. 


Sa en aes , 
By making = infinite, or the denominator y°—48a7y, in the ex- 
pression for it equal to zero, we shall find 
y = Oandy = +,/(48a’) : 


these results inform us of nothing new ; they belong to the point 
Al, which has already been remarked, and to the points F’, H, and 
I, which are not points of inflexion, but merely the limits of the 
curve in the direction of the abscisse. 

If we consider collectively all that precedes, we see that the 
form of the proposed curve is successively determined by the cir- 
cumstances presented by the points 4, D, F, J, H, K, and the in- 
finite branches X and X’. 
‘126. We shall conclude this Section by giving a fee practical 
examples, in the application of the preceding theory to the deter- 
mination of points of inflexion, multiple points, &c. 

Ex. 1. Let the equagion of the curve be 


ay me a", 


THE DIFFERENTIAL CALCULUS. 125 


This curve, which is a cubic parabola, has a point ot = injlexion or 
contrary oe at the origin of the co-ordinates. 
a”(ax — a") 


3a ; : 
If 2, = i and therefore, y = + a there is a point of in- 


flexion corresponding to each of these two points in the two si- 


, milar branches of the curve 
; Ex. 3. Lety? ae eE wom FE 
3d a a 
” ifs = rie and y = + v= there is a point of inflexion 
corresponding to each of these points. . ‘ 
» Ex. 4. Let «°—azry—b*y = 0. 
A point of inflexion, when « =U and y = 0. 
g Ex. 5, Let ar*-+by?+-ct = 0. | 

Two points of inflexion, one Pp eresportdini to « = 0, and 
y == —c Nae and the other to x img V- and y = 0. 

Ex. 6. Let xi —a*a*--a®y = 0. 

There is a point of inflexion pie to each of the points 

P _ 5a 
y inaki -z —. theref =: 
determined by making « = : 6 and therefore y ce 

Ex. 7, Let ay = 2% 

There is a point of dowble inflexion at the origin of the co-or- 
dinates, which is termed, as has already been observed (117), a 
point of serpentement or undulation, and may be considered as 
arising from the union of four points in the curve, 

It will aid the student in conceiving the nature of this point, to 
consider the curve whose equation is 

a’y = x —(b?-+-c?) 22-62 ¢? 

If we make 6 and c equal to nothing, the equation will coincide © 
with that given above, and the four points corresponding to « = b, 
2 = —b,« =c, andz =—c, will unite into one, which is a 
point of undulation. 

Ex. 8. Let ay?—a? —ba* = 0. 

rs d REE 
_ Atthe origin of the co-ordinates, we find <2 = af a which in- 
dicates a node or double point. 

If b = 0, the curve becomes the semidiltonn parabola ; in this 
case, the two tangents coincide with each other and the node is 
changed into a pornt of rebroussement or cust of the first kind, 

.- 


Se es 


125 THE DIFFERENTIAL CALCULUS. 


If b be negative, the values of = become imaginary, and this 


point is an insulated: or conjugate point. 
Ex. 9. Let 2*—ay2z?+-by? = 0. | 
A triple point at the origin of the co-ordinates, arising from the 


. i 

intersection of three branches of the curve: the values of = at 

ee b : 
this point are ++ Jf [ and 0. 7 

Ex. 10. Let x*+-y'—2ay°+-2b2°y =0. (Fig. 25). <A triple 
point or double node at the origin of the co-ordinates, 

If 6 = 0, the inferior ovals disappear and vanish ina point -.4, 
which may be considered as an invisible triple point. 

If a = 0, the superior oval disappears, and the point Ais a triple 
pomtl, arising from the coincidence of two tangents and a_ third 
tangent coincident with the axis. 


Ex. 11. Let y = 6+- (x—a)®. : 
tf « =aand y = 8, there is a cusp of the first kind. 


§ 
Ex. 12. Let y = 6-+-c2°+-(4—a)z 
if « = a and y = b-+-ca2, there is a cusp of the second kind. 


SECTION XII. 


Osculating Curves or the Curvature of Curve Lines. 


» , 127. It is by considering the relation which a curve:bears to its 
tangent, that the method of determining the various circumstances 
of its course has been learnt. Geometers,jhowever, have not re- 
strained themselves to this comparison of curves with right lines, 
from which they immediately separate themselves : they have pro- 
posed to themselves the’ investigation of those curves (among 
those of the most simple kind, as the parabola, the circle,*&c.) 
which, within a given small Space, approach nearest to any given 

curve. yin * 

The tangent of a curve being the limit of all the right lines 

which meet a curve in two points, we are led, by analogy, to seek 


THE DIFFERENTIAL CALCULUS. 127 


in general among all lines of a given species, the limit of those 
which cut the curve in any given number of points. 

We know, for instance, that to determine a circle requires three 
‘points ; we may suppose now, that they are taken in the proposed 
curve, and inquire, what circle we shall obtain on the supposition _ 
that these three points become coincident. This circle, called the 
osculating circle, will be the limit of all the others, in the same 
manner as the tangent-is the limit of all the secants. 

The latter line is determined by the two constants which enter 
into its equation ; and the circle by the three constants, which ex- 
press the abscissa and ordinate of its centre, and the length of its 
radius. 

128. It is plain, that when any two curves DX, EY, have three 
common points, M, MM’, M”, (Fig. 26) they will necessarily have 
three common ordinates ; and the lines P/M, M’O, and W'S, (Fig. 
10) have the same values in each. Denoting always, therefore, 
by a, y, the co-ordinates of the particular point WM, of the proposed 
curve DX, (Fig. 26); and by 2’, y’, those of any point whatever 
of the curve EY, we shall have, for the points M, M’, M", 


ie daeinif yy 
dy. WF dy. | dy FY 
age &e. a = &e. t is 47 at? &c. Par > &e. 
am ‘ | | diy ny OY 
sree te & Cc. re l dee &e — pera &e 
on the supposition that x’ is changed to 2 in the expressions of 4’ 
, F 2 U 
Se a 8 Sc. deduced from the equation ofthe curve EY. Now, 


if we pass to the limit, by making z = 0, the three intersections 
will unite in one point of contact, in which we find the oHlowing 
conditions must hold : 


y =y; 
dy _ dy 
dz’ dx’ 

d2y —s d?y 
dx? dx?" 


{f the curve EY be the circle represented by the equation 
(x —a)?4-(y'- 6)? = 77 
differentiating twice successively, we get 


128 THE DIFFERENTIAL CALCULUS. 


yay 
(2! —2) +(y “ae - 


Eat” nip ra = 0, 
and Supposing that 2’ is "onaagel to «, in these equations, they 
must then give the same values for y, a ai as in the proposed 
curve ; that is, they must be satisfied by the substitution of x, y, 
dy d? 


Ua, ere? 
a ee inthem, Making this last substitution, they become 


(am ag =e)? ots Js 4 1, 
; | 
iy ae ae ey Foti (2) 


dy? |, 
tas (y= Py whe, 3 (3) ? 


mo 


but since the quantities derived from the proposed curve are al- 
ready determined, by the condition of their corresponding to the 
particular point JV, it follows that «, 6, and 7, must have values 
assigned to them, proper for verifying these equations. 

From the last equation we derive 


dy? 
( za) 


Y= B= — diy a Eee ares ol 


Ee ee Se 


If in equation (1) we substitute these values of y— 6 and of « 
==, we Shall have 


(4 tae) (A a) dye. 


1 d2yy? (Te “dz2 sy? 2 QF 7 
(aa) dx2 J , ; 


and, by collecting the terms, we shall have 


aes 


tHE DIFFERENTIAL CALCOLUS, _ fag 


et” ae +5 
wi (: ar, ( (sae) "i 
dx? 


this equation may be reduced to 


Gt ye 
fdtyy? eS 


dx? 


and, by extracting the square root, it gives 


3 
dy2\2 P 
pa (ta) etanh 
Pag ~~ dad? y 
dx? 


129. ‘The circle whose magnitude and position we have deter- 

mined, varies for every point in the curve, since the quantities «, f, 
and +, on which these depend, are functions of 2 andy. It pos- 
sesses remarkable properties, discoverable either by geometrical or 
analytical considerations.. We shall begin by explaining the for- 
mer. 
Let MM M" M”, &c. (Fig. 27) be the polygon inscribed in the 
proposed curve. The circle which passes through the three points 
M, M’, M’,has its centre situated in the intersection of the right 
lines NO and .V’O, erected perpendicularly at the middle points of 
the lines MM’, and MM". If with.the points JV’, MM’, we com- 
bine a fourth MM”, we shall, by these three points, determine 
a new circle, whose centre will be in O', at the intersection of the 
perpendiculars VO’ and .V” 0", drawn from the middle points of M 
M’ and M’M". Conceiving now the same operate continued 
throughout all the extent of the polygon MM M'M”", &c. the cen- 
tres of all the successive circles will form, when Haine a polygon 
such, that all its sides, when produced, will’ meet those of the first 
at right angles. 

When we consider the limits, that is, when we substitute curves 
for polygons, the points M, M’,M’ becoming coincident, the right 
line NO becomes a normal to the curve, whichis the limit of the 


‘ 


polygon M M'M'A", &c, and a tangent to that which is the i- 


130 THE DIFFERENTIAL CALCULUS: 


* 


mit of the polygon O 0'0’0", Py and the circle which passes 
through the points M, M’,M", becomes the osculating circle. 

We must substitute then the (Fig. 28), instead of (Fig. 27), so as 
to replace the polygons by the curves DX and FZ, the second 
being the locus of the centres of all the osculating Eye of the 
first, which have the tangent JVO for their radius. 

130. In order to exhibit the analysis of the preceding proper- 
ties, we assume the three equations {1), (2), and (3) of art. (128), 


and clearing the last two of the differentials, which enter as divisors 
into them, we have 


(c—a)2-+(y— A)? Sy wee es 
(n—«) dx+(y—B)dy = 0. ec «gia (Bd: 
da? +-dy?-+-(y—p) d2y=0.% . . . (9); 


Now, Ist. since the second equation gives 
| dG yitin 
p-y=-F (e—2), 


it is (99) that of the normal drawn from the point whose ¢o-or- 
dinates are «, @, that is, from the point O of the curve FZ, to the 
point JM of the proposed curve DX, 


2dly. Differentiating the first two equations, not only with. res- 


pect to x, y, but also to the quantities «, 6, y, (inasmuch as these 
last are functions of the others), we get 


(cma@)de+(y—f)dy— (x —«)de— (y—p)dp=ydy, 
dx?+-dy? +-(y—)d'y—dadz—d@dy = 0. 
Now the equations (8) and (9) reduce these to 
~(2—4) dam(yp) dg = ydy . + (10), 
<da de—dady =0, . 


the latter of which gives 


dg ae. OF , 
an expression which changes the equation i 
7 da et 
: Bey wr (%— x) . 
{ A ess i “ha e 
into ? q 
“ *, 


ee ERI GAY: 


TER 
= 
ve 


& THE DIFFERENTIAL CALCULUS. 131 


P) of 2 dp 
y= B= da (x—a), : 
bab 


and which shows therefore (99), that the normal MO is a tangent 
to the curve whose co-ordinates are a, B, that is, to the curve FZ. 


3rdly. If we eliminate x —a, y—B, a between the equa- 


tions (7), (8), (10) and (11), we shall have es 


dy! = det +d8?, aN ji4 te, 


which gives the differential coefficient of vy, with respect. to the 
variable #: now (105), this expression is also that of the differen- 
tial coefficient of the arc of the curve, whose co-ordinates, are «, 
8 ; and it follows, from this identity, that the radius of the os- 
culating circle varies. by the same differences as the are of the 
curve f'Z, a property which merits the greatest attention. 

In fact, the radius of the osculating circle at the point VM, being 
a tangent to the curve FZ, has its direction necessarily the same 
with that of a thread wrapped round the convexity of this curve, 
and then unwound as far as the point O. We may observe, if we 
trace this development in its progress from O to O’, that the thread 
increases in length by a part equal to OO,’ the arc of the curve 
FZ ; and since, by what we have before said, the difference of the 
radii OM and O' M’, is also equal to the same arc OO’, it follows 
that the extremity M of the thread must be still found in WW’, a 
point. in the proposed curve, which it has never quitted during the 
progress of the development from one of these points to the other. 
We may therefore regard the curve DX as generated by the de- 
velopment of FZ. 

This process | has much analogy with the description of a circle. 
The curve FZ performs the part of a centre, andthe radius MO, 
instead of bemg constant, varies at every point. FZ is called the 
evolute, the curve DX its involute, and the radius of the osculating 
circle, the radius of curvature. 

In general, the osculating circle at once touches and cuts the 
curve, in the manner of atangent at a point of inflexion If 
the radius of the osculating circle increase from to JM’ , itis 
evident that the arc MM’ of the curve must lié above GH, the 
osculating circle of the point J, while the part MD lies below it. 
Moreover, since we may always conceive the points Mand M’ so 
near each other, that the radii MO, M’O' may differ by any quan- 


aN 


al 


, PA RS wu te Pot ya RS aS ) fie 
* * dae eS Bets: ye) eM eae 


132 THE DIFFERENTIAL CALCULUS. 


tity, however small ; and since, if\we describe the are G'MH’ with 
the radius Mo = MO, the arc DM will be entirely below the curve, 
we shall easily perceive that no circle can pass between a curve 
and its osculating circle; for every circle, whose radius is less 
than JMO, will pass entirely within the arc GMH, while every circle, 
whose radius is greater than Mo, will be entirely without the are 
GMH. 

The osculating circle being, therefore, that which of all circles 
touching the proposed ¢urve at the point. JM, approaches nearest to 
it, on either side of the'point of contact, is consequently that which 
differs the least from the curve at the point under consideration. 
The curvature of a circle is uniformly the same in every point of 
it; but inares of a given length, that of a smaller circle is greater 
than that of a larger, so that the curvatures of these arcs are in 
the inverse ratio of the radii of the circles to which they belong. 

‘We may, therefore, by the radius of the osculating circle, es- 
timate the curvature of the curve at any point. This is the reason 
for calling the radius of that circle the radius of curvature : and 
it appears that the curvature of any curve is in the inverse ratio. of 
the radius of curvature. 

The evolute may also be dénsidered as the limit of the inter- 
sections of the normals of the proposed curve, taken two and two 
consecutively, since the point K, the intersection of the two radii 
MO and M'O', perpendicular to the curve DX, at Mand, ap- 
proaches so adel the nearer to the curve FZ, as the points M and 
MM are nearer to each other. a 

131. We may likewise prove by the’assistance of analysis, that 
between the curve proposed, and its cirele of curvature, no other 
circle whatever can pass ; and this propery leads us caine e 
to the others, 

In general, when two ih hy whose ordinates and afkasteielice! are 
designated by x and Ys x and y', have a common point, and in 
which consequently x = 2, y = y, if we > take the difference of 


the series © ah? q 
dy i ay 2 d3y @ 
ee et ae 1.3.3 1? Se ’ 
dy iy @ dye : 
stage; Me 2 ' dzx'31.2.3. tas t 8. R F 


which express the ordinates of the points corresponding tothe ~— 
abscissa «-+7, we sr find, erapally | 
ty, 


Ps 


> 


f 


«i DIFFERENTIAL CALCULUS. 133 


diy? | dy 
(2-2 rt dx”, a dx? a) ros 2. 3+, &e. 


for the expression for the distance of these curves, in the direction 
of the ordinate ; but if at the particular point, which we are now 


dy’ . dy 
shies we have a= = ae this distance will be then reduced to 


days” as dy a3y’ ° ’ 
dz? iat aie ie) Tepe 


The ratio which this development bears to the preceding, becom- 
ing smaller i in proportion as 7 increases, it results from it, that 
the distance which it expresses between the two curves, will termi- 
nate by being less than that which is expressed by the former ; and 


that consequently no curve whatever, for which we have not 
dy d 
a= == , can pass between those which Satisfy this condition. It 


is on this account, that between a curve and its tangent, we can 
draw no straight line passing through their point of contact. 
The proximity would become still greater, if we also had 


ata dty 
dx’? dx?" 


A curve which only satisfies the two conditions 


that is to say, which has only a simple contact, cannot approach 


at those parts which are indefinitely near the common point, so. 


near the second curve as the first approaches, and cannot possibly 
be drawn so as to pass between them. This is the case of a circle 
which is merely | a tangent, when compared with the circle of cur- 
vature. 

ify = y, Ad = = cy ae aa then, since the first term of 
the expression for the difference of the ordinates involves 2’, which 
changes its sign when we substitute —7 in the place of +2, we 
readily see, that the circumstances which we have remarked of the 
tangent at points of inflexion will likewise take place in the circle of 
curvature, excepting those cases only in which we have 


¥ 


ay 


ae 


is4 THE DIFFERENT CANCULUS. 

It is not necessary to extend these considerations farther, in or- 
der to be convinced, that curves may have with each other degrees 
of contact more or less immediate, By pursuing the course aiadi- 
cated in art. Liee), we should find, that if two curves had four points 
in common, and if we would determine one of them in such a 


manner, that these points might coincide, we must then have, at 
the same time, 


dy) Nay day 'e d2y" dey" ndey 
ham. ax’ Pada as dx? dz?’ dx8 dx?’ 


This contact would differ from the preceding in this circumstance, 
that a curve which has with either of the proposed curves,. a con- 
tact of the above-mentioned species only, cannot be drawn between 
this and the other. . 

By employing the preceding conditions, in the determination of 
the constants which particularize the equation whose variables are 
x and y, we shall discover, that this equation must necessarily in- 
volve four constants. 

132.. We divide contacts into different orders, according to the 
number of points of intersection which are supposed to be united 
in them; or, what amounts to the.same thing, according to the 
number of terms which are supposed to be equal in the develop- 
ments of the ordinates relative to a consecutive point. The contact 
of the highest order which can take place between the tangent 
curve, and the one proposed, depends on the number of constants 
which the equation of the former poles; and is also called the 
contact of osculation. 

Thus the tangent, between which and a given curve, a simple 
contact only can take place, is an osculating line of the first order : 
the circle whose equation involves three constants, may have either 
a simple contact of the first order, or a contact of ;the second; but 
this last, being the most elevated, is termed that of ‘osculation, and 
distinguishes the circle of curvature from all those circles which 
are merely tangents. in 

133. We shall not detain ourselves long with the application of 
the formule 


. lae-tay)! 
 dad’y ” 


dy. dy (dx? dy’) 
; Te PO day 


a if, 
» ‘ 
THE DIFFERENTIAL CALCULUS. . 185 
, ath dx? dy? 
Ym —=— = ————— * 
: J gay 


since they can present no. difficulty when we are well acquainted 
with the structure of the differential calculus. 

The value of y being susceptible of the double =, it may be 
asked, which of the two we ought to employ; for it is very 
evident, that in general for one point of the curve there is but 
one radius of curvature; and since this radius has not, except 
at some particular points, the same direction with the ordinate 
or abscissa, it cannot properly be said to have any peculiar 
sign, with respect to those lines. The determination, there- 
fore, of that by which we commonly affect it, must depend 
upon a convention previously established on the direction of 
the curvature, with respect to the normal. If we agree to 
consider the radius of curvature as positive, forthose curves whose 
concavities are turned towards the axis of the abscissz, as the value 


Bye a 
of —? is, in these cases, negative (108), we must affect the ex- 


pression for y withthe sign—; and the same assumption will also 


make the radius of curvature negative, when the concavities of the 
curves are turned from the axis of the abscisse, since it changes 


2 


Pris ay : , 
its sign at the same time with om In conformity to this con- 


vention, we shall always, in the application of the formule, as- 
sume 


_ (dattedy?)? 
ue dad?y ° 


134. In order to give an application of this formula, let us find 
the radius of curvature of lines of the second order, whose general 
equation is, 8 | 


2= ma-bne’. 


_ By differentiation, we have 


(m+2nx)dx 
a 
y Qy 


9 


Qnyda*—(m-+-2nx)dedy __ § Any? — (m--2nsx)?} da? 
ee ea aa ee 
y 
there will thence result 


: 


o, 


136 | THE DIFFRREN TIAL CALCULUS: 
ce 
ae y?-+(m4-2ne)?}* 
Sny? — = 2(m-p onc)? ; 


If we substitute the value of ¥?, in this expression, we Aria 9 ait 


§ 4( patna nx? ) )-b (mF 2na Qna)? yee 


Om? 


This is the general expression for the radius of curvature, in lines 
of the second order: we shall deduce its particular value for each 
species of these lines by giving to mand n the’values vee re- 
spectively correspond to pe 


This result is reduced to “ in all cases, when 2 = 0; the cult 


vature, therefore, of the proposed lines at their vertex, is the same 
as that of a circle described with a radius equal to the semiparame- 
ter. , ¥ 

¥n the parabola, in which n = 0, we have simply 


m2 ie 
f fy cmemieese 3) sau ; 
v “y ame i (m?+-4mx)? , 
5 = 2 . 


2m" ° 


‘Now, since the normal of the parabola has for its expression 


2 
J (Gt); 

we see that the radvus of curvature of the parabola 7s equal to the 
cube of the normal, divided by the square of the semiparameter. 

135. We may apply, in a similar manner, the general expression 
for «—a and y—£ ; and substituting for y its value, we should 
have two equations in terms of x, «, and 6 ; from which, by elimi- 
nating x, we deduce the equation of the evolute, in terms of « and 


B alone. We shall go through this process for the parabola only. 
We have, in this case, 


and there results, 


ay? sy? == iy. 
TE ne (aie Ay Ty 


Ae 


aes ¥. 


= age “i ‘ 
THE DIFFERENTIAL al 137 


4y?--m? 


2m 


Dis « 2 
Fg gh om dy? 44? es pth 


2y m2 4y2 


from which we get 


hapa Me 2y? | 
ig ehh AT Ta | 
substituting, in each of these equations, for y its value ./(mz), 
there will arise 
4a? 
—B = ys li = — 2x — im; , 

determining the value of x, in the second result, in order to substi+ 
tute it in the first, we shall obtain 


& =A (e—im), ~im)° + 

the last of these equations belongs to the evolute of the parabola. 
If we change « — 1 into @’, or transfer the origin of the abseis- 
se to D, (Fig. 28), we shall be able to give it this very simple 
form, 


5. 160! 
Bult 27m ’ 

which shows that the curve DF is a parabola of the third order ;* 
composed of the two branches DF and Df, the first of which gene~ 
rates by its development, the branch 4X of the common parabola 
XAzx, and the second produces the branch Az. 

It is easy to prove that the branches DF, Df are convex towards 
the axis of the abscissw: for by differentiating the equation 


baal (ae) Xue on = nel 


where n = a/ ( ie =—)s' we shall find 


d’s 3 
da ag Af at 5) 


ie 


* The equation ye == ma, being generalized thus, ve ine, represents a farnily of 
curves of which the common parabola is only a particular case : they are alsonamed 
parabolas, but they are distinguished by the exponent of the degree of their respective 
equations, ' 

T 


128 THE DIFFERENTIAL CALCULUS. 
apositive value for the second differential coefficient ; therefore 
the branch Df is convex towards the axis of the abscisse, when 6 
is positive ; and, when # is negative, the second differential co- 
efficient wil! have the same sign ; it likewise follows that the branch 
DF isalsoconvex towards the axs of the abscisse. 

_ 186. It is necessary to observe, that in order to describe the 
parabola XAr, by the development of the curve FDf, the string 
which is wrapped round one or other of the branches DF and Df; 
ought to have at the point D, in the prolongation of the tangent 
BD, alength AD, equal to the radius of curvature at the point 4; 
that is to say, equal to half the parameter of the given curve ; 
every other point J, taken upon this string, would generate a dif- 
ferent curve. If the point J should fall upon the point D, the radius 
of curvature of the curve described in that case, would be equal to 
nothing at its origin, and Spreervenhy the curve would have, at 
this point, an infinite curvature. 

Since the arc DF is equal to the difference fret eden the radius of 
curvature MF, corresponding to the point JM, and the radius AD, 
which belongs to the origin of the abscisse, we easily see that the 


curve FDf is rectifiable ; that is to say, that we can assign a right 


line which is equal to it in length. 

This remark is general; for since we can always deduce an 
expression for the radius of curvature of ,algebraic curves, the evo- 
lutes of these curves are all rectifiable. 

137. We shall now conclude this section with a few practical 
examples. 


b2 ‘ 2 g : 
Ex. 1. Let y? = — eke —x?) or “ ahs = f. 


If we make a y aos aig? » we ae the radius of the circle of 


curvature, or 


3 
és (a? — ex?) * 
LAiciys a: 


if # and B be the co-ordinates of the centre of the circle of 
curvature, we have 


DY il. laden tae. FAB 
" i ai 


» oe 


tn, 


THE DIFFERENTIAL CALCULUS. 139 


a et 
« = and 6 = 7’, the equation of the evolute becomes 


) NV 2 + 
© 8 ©y'+@)=1 
a ‘ 


In the case of the hyperbola the equation of the evolute is 8% 


a \5 Y Nie 

G)'- ("= 

a2 2 az 2 

where @ = ee and 6° = AA 
a 

Ex. 2. Let zy = a%, or let the curve be the rectapaiie hy- 


perbola referred to its asymptotes. 


9 on) a3 f at 2 
ke a aa G %: ro ae 
& 
3a as 3a? uF 
“= “gare  aatt 29a 
: ee ‘ ‘ 
In order to determine the equation of the evolute, we find | 
| : age 


a tee! C4) f= 504)" 


oom foe CoD) f=3G-3) 4 


2 


consequently ~ 


203 ‘ 
AV 5 = e+e) +¥ (=8) 


aE 5 = Viets) — Vie-P) 


a 
and therefore, 
2 2 2. 
(4a)° oo (2+8)5 — (#—£)%. 
Ex. 3. Let a*y = 2, the equation of the cubical parabola : 


ee (at-+90%)% 
“ we 6a°x 
This expression is a minimmum, or the curvature is, the greatest 
when. 
* | A BI + ‘ ® ig 
. 


_ ~— 
ay 
~ i 
~ 


440 SHE DIFFERENTIAL CALCULUS. 


a inti a 
Me 4/45, 47 OT1 88): 
i, Ss GP ae Bx3 
Also «a = Q 8 gt? p= reser hy 
‘The elimination of x from these equations, leads toa very com-— 
plicated expression for the equation of the evolute, which does not 


seem to admit of reduction.* 
Ex. 4. Let ay? = «°, which is the equation to the semicubical 


parabola : 


11 

Aa?a? 9x, 2, 

y Suiio BEN eae 

If x =0, y = 0, or the curvature at the vertex is infinite. 
| gil 9x? . 
ihe Panes Di 2a’ 
bake x 

e) = Seta 
Ex. 5. Lety = AG 2 ‘ 
§ at-++-4aa°— 4a YF 


ae er a*( 3a — ~ 4ey 6 
If «<=: 0, y =@: at this point y = m, and the curve coincides 
avith its asymptote. 
3a : : ; : 
Iifs= ah ee, which corresponds to a point of inflexion 


_ a 10a2— 1224 | 
227(3a—4x) ° 


AY (ar — x ; Aasige ) 
B= =) des a ie —o'2-+-ar—x . 


* The only difficulty attending the determination of the equation of the evolute of 
an algebraical curve, arises from the elimination of x from the expressions for aand £, 
which generally requires the meagre of equations of higher orders. 

¢ SR 


ry 


! , Oh i; eae 4 | 
ud - Sta (ars i Se 


THE DIFFERENTIAL CALCULUS. 141 


SECTION XIII. 
The Differentiation of Functions of two or more variables. 


138. When we have one equation between three variables, we 
must first arbitrarily assign the values of any two of these, in order 
to determine the third, which consequently is a function of the first 
two. If we have, for example, the equation 


x+y? -+2? = a, 
we cannot obtain z, without first having assigned values to x and 
y ; but it is proper to observe, that the quantities x and y, having 
no relation established between them, the second may remain con- 
stant, although the first should change, and reciprocally. 

It follows from hence, that the value of z may vary in several 
ways ; Ist, in consequence of a change in the value of x ory ; 
2dly, by the coincidence of both these circumstances. In the 
first case, the quantity y or the quantity x, being considered as con- 


stant, the proposed equation is, in fact, an equation of two varia- 
bles: thus, when x alone changes, we have 


xdzu-+-zdz = 0; whence x-+2z ’ cai), 


and when y alone changes, it becomes 
: dz 
ydy+zdz == 0; whence ytza, == Q, 


We have then, successively 


but it must be observed, that the first of these equations is relative 
to the particular variation of x, and the second is relative to that 
of y, which is usually expressed, as has been observed (66), by 
saying, that one is the partial differential relative to 7, and the other 
the partial differential relative to y. 

The meaning of the proposed question will prevent us from con- 
founding them, and they are otherwise sufficiently distinguished, 


142 THE DIFFERENTIAL CALCULUS. 


by observing the differential of the independent variable, which 
affects hans” e 
The analogous differential coefficients are 


tie. 


In general, when a function of-several variables is concerned, it 


should be remembered, that in a dz is the partial differential of z, 


relative to 2, whilst in Me : 

139. If f (x, y) represent any function of « and y; supposing 
at first, that the variable x alone changes, and becomes x~+-2, we 
must consider y as a constant quantity, and treat the proposed 
functionin the same manner as a function of a ; we shall therefore 
have, putting i ae ig (, Y), 


dz is the partial differential relative to y. 


du 2? d3u 7° 
= u A a ae G | 
FS (a-+t, y) ut = ito 2" da? ‘og Ft eee Kae 
In order to find what the proposed function becomes when y alone. 
receives an increment k, we must consider x as constant, and f(x, 4), 
oru, asa function of y; from which we have 
2 3 
d2u k2 ik ee ee 


io} se 
fle y+h) wt$ “aires 2 ans 23 


In case the two quantities x and y vary at the same time, and be- 
come x-+-2 and y-+k, as we have not assigned any particular form 
to the function f(a, y), it is not possible to make the two substitu- 
tions indicated at the same time ; but itis easy to perceive, that 
we shall arrive at the same result, by first putting «+7 for x, and 
afterwards writing y+-k for y, in the development obtained by the 
first substitution. 

We have already found the development of f(2+-7, y) supposing 
a alone changes. In order to develope the coefficient of the diffe- 
rent terms of this series (1), having regard to the change which y un- 
dergoes, we may first observe, that in each of therm « may be consi- 
dered as a constant quantity, and that we may consequently treat 
them as functions of y alone. According to this, the substitution of 
y-+-k instead of y, will change, i in the dévelopméli above ajtudes to, 


oa d?uk2 . d3y ks baat th 
i} w into alg ay tage 3 yO ays "2a 33 —f- ei ibs: 


i 


THE DIFFERENTIAL CALCULUS. 145 
du du 
E aie Wel Men d3,— 
du. dw’ “dz i ke ios. k8 
: ‘ i into Rta rer ag aye ee , &c. 
» d’u d2u, 4 du 
d?u. d2u_ ‘dz? “dx? k2 da? ks 
ery into Gat ye “aya! ety “oun Xe. 
# &e. &c. Xe. &e. &e. 
du 
a) 


indicates two differentiations 


But since the expression 4 : 
ae 
made successively onthe function u, in the first considering x 
alone as variable, and in the second making y only vary ; this ex- 
d24 
dyda:: 


pression assumes a more simple form, when written thus : 


ae) ivy 
agile is represented by dy? 


t 


In the same manner, uty and gene- 


rally, by 5 , Is meant the dmettatial coefficient of the nth or- 
— dgrdr™ 


_ aan he ‘ _—. Ary 
der, of the function * ay supposing that y is the only variable in it, 
whilst that function is itself the differential coefficient of tHetor- 
der m of the proposed function, supposing x ih " ge 


140. This being premised, let the Wlaes ore, 22 n —, ee ; 8, found 


above, be sibstituted in the development (1) of f(x-+2, y), and by 
arranging it so that all those terms in which the sum of the expo- 

_ nents of 2 andk are ge shall be in the same vertical column, 
we shall have 


d2y, k2 
abi yth) = ut oh og ty Be = 
P jth the 
2 d?u 42 
i Sathndg i i 
gs a agi | 
+ | a +, &e. J 


ee his development has been obtained by first putting x-+2 
of a, and then yk instead of y ; but we might have pro- 


me. 
ys 


3 


i144 THE DIFFERENTIAL CALCULUS. 


ceeded in an inverse order, and have begun by the substitution of 
yt+k for y, and then, by the substitution of ai for « in the deve- 
lopment, (2), thus obtained ; we should have ae by proceeding 


as in the last article, 


fleti, yt) ut S452 S 4, &e. | 


d2 
+a i+ 5 ik+, So. | 
(t). 
dzy k2 
+ — dy oa me &e. | 
+; &e. J 


It is evident, that this development ought to be identical with the 
former ; since it is indifferent, whether we first change x into 
x-+-2, and then y into y-+-k, or whether we make these substitutions 
in an inverse order, as in either case, we obtain f(2-++-7, y+k), 

If we compare, in these two developments (3) and (4), those 
terms which are affected with the same powers of 7 and k, we 
shall find the following series of equations : 


a aa " 
a ‘dydz — ddy 
Gu at 

dydx? du? dy 
By! Cy 


dyedz — dudy? 


dntmy, dmtny > 
dynda™ dx dy ; | 
From the first of these it follows, that the differential coefficient 
of the second order of any function of two variables, taken first 
with respect to one of these vartables, and then with respect to the 
other, is the same im whatever order the differentiations are per- 
formed. 
Take for example, we i 
‘i langage Mo meh (ety?) 2 
Qx2 27 pts : ete 


ee 


g 
THE DIFFERENTIAL CALUULUS, 145 
if we first differentiate it, considering x only as variable, we have’ 
dy 2? Fy? --y4/(x? +9) 
———- . a ae ae re 
a: = 


then, differentiating this resulf, with respect to y only, we obtain 


d2u aly Qy FJ (2? Fy’) ye 


P dxdy C wi / (a? y?) © 


and, by performing these operations in an inverse order, we find 


du —=y—4/(u2-+y?2) , 1 
=v te 


dy x? 
Pu yt Vv (etty) 
dydz x (ay? 


And by reduction, we shall find 
; fu 4 duh 2y a? Dy? 


dxdy  dydx x * x? 4/ (Ly) 
The theorem expressed by the equation 


| du. du 
dady  dydx” 


is of great importance in the integration of differential equatioug 
of the first order and first degree, involving two variables, as it 
enables us to ascertain whether they result from the simple differ- 
entiation of a function of x and y, or from the elimination of acon- 
stant from the equation in which they are involved. 

Again, let wu = x sin y+y sin x: 


du : 
= = sin y+ y COS &, 


pai, 
sey = C08 O08 2s ‘ 
- = an ua COBY 
du 


~—— = Cos x - Cosy; 


du 43 d?u 


es dada: es = cos x-+ Cos y. 


U 


146 THE DIFFERENTIAL CALCULUS. 


The reader may make use of this or the last example, in illustration 
of the theorems, 


du * du ee 
dad — dytds ~ dydedy’ 
dtul dtu day 
iat) aR wybelias 
SS dt ie dtu ie ae 
~ dnalyelaaly  Gaayede  dyeedy? 
&ce. &c. 


142. Subtracting f(x, i or & from f(x--i, y-Fk), we find 


d24% 2 
AEH 9th) flg) = ee 
aka 9 odtead gle | 
7% dy add ik-+-, &c t ea 


If we extend the definition (1'7) of the differential of a function of 
one variable to those of two variables, we shall perceive, that the dif- 
ferential of f (x, y), or of u, consists of two terms, which form the 
first column of the preceding development, and by changing 7 into 
dx and k into dy, we have . 


du, .d 
d.f(x,y) = du = Felt t Gt 


From this it follows, that the complete differential of two varia- 
bles, as has already been observed (66), consists of two parts, 


. du 4 
that is, ant or the differential taken on the supposition that x 


; di 
alone is variable, and 7%? or the differential taken when y only is 


variable. We may, therefore, apply to functions of two variables 
the rules which have been given (Art. 19, 20, 21, &c.) for those 
which depend only on.one ; and for this purpose we must differen- 
tiate the given function, first with respect to one of the variables, 
and then with respect to the other ; aud the sum of the two results 
will be the complete differential required, 

143. It will not be necessary to give many examples relative to 


THE DIFFERENTIAL CALCULUS. 147 


the differentiation of functions of two variables, since it is redu- 
cible to that of functions which contain only one: the following 
will be sufficient. 


Ex. 1. Letu = xy": we have 


onde = ma™—lyndz, 
du 

— ) m7, n—t1 e e 
dy! = nx™y"—"dy 3 .*. 


du == ma™—y"da--nam y"—Idy = a™—1y"—! (mydx--nady). 


an 
Ex. 2. Let wu = — 
y 


du du 
dy = Bet Ry 


5 x"—l(nyda—mady ) 
. actor Sa Ties 
ng Ym 


Ex. 3. Let vw = arc. tan. ; which is the expression for the arc 
of a circle whose radius is unity, and tangent ; In order to dif- 


ferentiate this, we must make = z, and must find, from art. (44), 
4 


the differential of the arc whose tangent is expressed by z; the re- 


ai! Ge Ras eae 
sult is ibee : we have, therefore, du = ere and putting for z 
and dz their values, we shall have 
ydx—axdy 
aE Lahde 
du = HS bog aE 
a _ 
» ‘ doa | 
Rx. 4. Letu = eer | = ay(x?-+y*) * 5 
then we have a 
du. ayxda 
qa —— Bok 
(a? -+y") : 


OAM 


148 THE DIFFERENTIAL CALCULUS. 


ody Th sis TiS 7 ae j 
(tty) (arty)? 

therefore 

SUE Sh I ea AY ered 

(attyt)® (attye)E Gay)? 


du=— 


or by reduction . 
bis 2 
du = ee? dy | 
ity re es Pye 


Kx. 5. Letu = 1 tan: if z = —, we have 


du "cot eh 1 


° . dz cos  sinzcosz’ 


hs. but since 


therefore 
_ - ydn m edly 
, rte atc 
y* sin — cos — 
i ye 
144. The manner in which the differentials of functions, which 
depend on more than one variable, are written, gives rise to some 
important remarks. We have already seen (138), that, in this case, 
du 


7 dx must not be confounded with du, which it might be if « con- 


; : ated 
tained only the variable «, because the expression =, has, in the 


ease of wu being a function of two variables, a particular meaning ; 
¢ it denotes the differential coefficient taken on the hypothesis of « 
only being variable, or, it is the first term of the development of 
ihe difference taken on that hypothesis, divided by the increment 


ik. : 
dx; and _ signifies the same with respect to 7. 
yo it 
Wer, aud | 
The quantities hee 


dx’ dy are commonly called partial differences 


i 


~ 


i) 
THE DIFFERENTIAL CALCULUS. 149 


amin 


of the first order of the function w; and dl sengraly, damdjw ePre- 


sents one of those of the order m-+n, which arises by aad 
m times with respect tox, and n times with respect to y. 


It ought, however, to be observed, that the term partial difference P 


is not a correct one, for the formule which are thus denoted do not 
express the difference of two quantities. The real partial differ- 
ences of ware 


f (ert, y)—f (2s 9) 
Sf (% y+) —f (a; y)- 


es 
The first of these is taken, with respect only to the change of z, 
and the second by supposing y only to vary. The expressions 


du . du U du 
ee es dy k, or es dix, dy dy, wt, 
which are the first terms in the developments of these functions, 


ought to be called partial differentials, whilst dx + dy is the 


du. du / ; “3 
total differential ; and a =; are alwaysthe differential coefficients 


of the first order of the proposed function. It should, however, 
be observed that a function of one variable has only one differ- 
ential coefficient in each order, while a function of two variables 
has two differential coefficients for the first order, three for the 
second, &e. 

145. These coefficients may be deduced in the following manner, 
from the first‘two, beginning with 


du du 
ow mag d 


7 1 : ad a j " 
then taking the differentials of the functions “and ay Which must 


dx 
be treated as functions of two variables, we have 
du d?u d2u 
"2 dee OT dyda 
f du du d2u 
ee de 4 ly - 
dy  dxdy a dy? J 7 alt 


and since the Second diffaiiatial is nothing more aly the differ 
ential of the first, we shallhave — 


iad 


> = 
150 THE DIFFERENTIAL CALCULUS. 
iy 1 
wt d*u du 
im | OS a as, daly +o di ae 


. considering dx and dy as constant quantities, and observing that 
the differential coefficients whose denominators contain only the 
products of dx and dy differently arranged, are identical (140). 
If we differentiate the differential coefficients which occur in the 
‘ preceding result, we shall have 


d2y%. ..d%u diy 


| de> Tighe at dat dy, 
« | ft : =. = ine lege dy, 
Pa i . Ks iF s, : oe ao let TS dy, 
_ and sar 
q ee OY ai OEM. detidy + s = de pbs dy. 


This formation may as be continued, and the analogy of the 
results with the powers of a binomial will doubtless be observed. 

It may be noticed that the series, art. ( 142), is the same with 
that/in art, (54), when we substitute dx for 2 and dy for k; : so that, 
if we denote f (a-+2, yk) y u', we stillhave _ | 


Be sepitoe Siti 5. zat &e. | 


which formula is quite as general as that of ite, (149); ; since the 
increments dx and dy are arbitrary: 

146. Itis easy to extend these considerations to functions of 
any number of, variables, and to convince ourselves that if he 


have | 
u =f (t, 2, 4 2), ae ‘aa wy 
there will result | , , 
| | an ! 
du = oe dt dopo ik dz, 
re 
denoting by | 
‘ | fs 
; du du du du 
dt? dx dy’ dz’ 
® 


THE DIFFERENTIAL CALCULUS. 151 


the differential coefficients of the function w, taken on the suppo- 
sition that ¢ or x, or y or z, only varies. 

This notation, which owes its origin to Fontaine, is the most 
simple, and the most expressive of any which has yet been propos- 


ed to denote the same operations. Euler, fearing, lest the diffe- . 


rential coefficient = should be confounded with the ratio of the 
Hy 


complete differential vu, to the differential dz, which ratio is equi- 
valent to 


i rR, eS 


denoted this ratio by whilst he expresses the differential co- 


ficient by (=) . The nature of the subject almost always renders 


this distinction superfluous. Fontaine, however, had provided 
against any case where this might be absolutely necessary, by pro- 


, : 1 
posing that the ratio may be written thus : ad ; and knowing 


that this ratio occurs much less frequently than the differential co- 
efficient, he gave to this latter the most simple sign, which is conforma- 
ble to all nomenclatures, and is exactly contrary to what Euler did. 

Observing that we never employ the ratio of the differentials of 
two. quantities, without supposing, at least implicitly, that one of 
them is a lage of the other, and that the expression 


4 edt = det * dy t dz 


pre ee 


—+—_—_——,, or — du 
iz riage 
uk has no meaning, unless we consider the variables ¢, y, and z as 


implicitly depending on x ; Lacroix has proposed to write the ex- 
pression thus : 


d(u) 

“dx? 
enclosing the function between two parentheses, to show that not 
only alithe terms explicitly containing « may vary; but also all 
those quantities which may implicitly depend on it. ‘This notation 
has, as well as that of Fonrarne, the advantage of preserving the 
most simple sign for the case which most frequently occurs. 


SS 


152 THE DIFFERENTIAL CALCULUS. 


147. Let v= 0 be an equation, containing 2, y, and z; if we 
consider z and y as independent variables, z will be.a function of 
both ; and when a receives any increment, y being» supposed con- 
stant, z will undergo a corresponding change. On this hypothesis, 
the equation u = 0 ought to be considered as one equation between 
~ two variables, x and z; we shall have, therefore, 


du dudz _ 

di dzde 
and nae this may be nelneed the differential coefficient of z, re- 
lative to the variability of x. It must be remembered, according 


og: _ dz es 
to the distinction which has been made (138), that in Te dz is the 


partial differential of 2, taken with respect to the change of x alone. 

It is evident that if we made y vary by differentiating the pro- 
posed equation, as if it contained only y and z, we should have 
had | 


—_—+—.— = 0. 


If we multiply the first of these equations just found by dx, and the 
second by dy, and if we then add them together, there will result 


du du du {dz dz : 
—d cy ——§ — a — Orr 
oa belgie Nia e+ Fly) ae 


dz, dz, | ae 
but dent Tye is nothing more than the complete differential of 


z (142): we have, therefore 
du du du 


that is to say, we may equate the first differential of the equation 
«w == 0, taken with respect to the three variables x, y. and z to zero. 
Tt must not be forgotten, that this differential ought to be consider- 
ed as equivalent to two equations ; and by substituting in it instead 
of dz, its value 
tee 

those quantities which multiply dx, and dy, must, on account of the 
independence of the increments, be separately equal to zero. 

148. The equations which give the coefficients of the higher or- 
ders may be deduced by differentiating the equations 


P ?He DIFFERENTIAL CALCULUS, | "ges 


: s oa 8 oda ae 


dz "ee" ” dig o es (1),. ~ 


— du | Vy das 9 
' ad 


which, containing i in general the three variables, «, y and 2, ought 


dd hs Sie ee ae Po tye 


i 


Bt i 
Be a) 


to be treated in the same manner as the proposed equation. And. 


considering first the change which z undergoes, not only will z 


vary, but also its coefficient of the first order i which will give 


rise toa coefficient of the second order. Therefore, by differen- » 


tiating with respect to x, the equation (1) 9 we shall have, as in 
equations of two variables, 


d*u d2u dz is dz? , du dz 
dae |” dede'da def dat de det 
Differentiating afterwards, either equation (1), with respect to y, 
or equation (2), with respect to x, and observing that, in the first 
d?z ; Ger ane ad? 2 
aya’ and in the second, Fa gives = 
we shall have in both cases the same result, which will be ex- 
pressed by aN 
d?u  @u dz, d8u. dz. .Pudzdz, du dz . 


dz 
case, —— gives 


Sab eles 


Lastly, by differentiating the fapation @), with respect to y, and 


) dz 
observing that —, in this case gives ——, we shall have 


dy ig 
a2 u d?u dz vase dz? du d?z _ 
det dydz dy Lia ay de ae. 0 Dy a aie (5). 


The function z having three differential coefficients for the second 
order, each of these may be*determined separately by one of ad 
above equations, that is to say : 


= = by the equation (3), 


Pz 
4), 
daly By the Sicaiion ( 
d3z ! 4 
a 5), 2. yt 
ag by the equation (5) , 
x 


edly ~ aya? 


Fe 


154 THE DIFFERENTIAL CALCULUS. * 


since all the differential coefficients of the function w, which is 
explicit with respect to the'variables x, y and z, are given by the 
differentiations indicated. 7 

If we multiply respectively equation (3) by dx?, equation (4) by 
dady, equation (5) by, dy’, and that we add the nent together, and 
replacing the terms | Mu oe 


eats 7 ob by dz, 


Ps 
“d oe years <5 = dy’, by dz 


we shall find the same equation as that which we would have ob- 
tained if we had differentiated the equation 


du du du 

an gt dye dz =0; 

considering dz and dy as constant, and z as a function of x and y. 
149. These considerations may easily be extended to any order 

of differentiation, and to any number of variables ; for the whole 

consists in determining those which are independent, which can 

only be done from the nature of the question which led to the 

proposed equations ; and we must then differentiate with respect to 

each of these variables, treating the subordinate variables as im- 

plicit functions of the co. ones. If, for example, we haye 


the two equations, 


between the five variables, s. ¢t, 7, y, and z, we see that three 
of these variables are independent. Supposing then that y 
and z are functions of the other variables, s, ¢, 2, given by the pro- 
posed equations, we must successively differentiate u and z, with 
respect to s, with respect to ¢, and with respect to «; and we shall 
have 


du da gud dy , du dz 
ds dy ds ' dz ds 
du du dy du dz _ 
dt ‘dydt dzdt” ” 
. du dy , du dz je? 
dy da‘ dzdxz 


= 0, 


“ee 


* 


THE DIFFERENTIAL CALCULUS. 155 


if we multiply each of these equations respectively by ds, dt, dx, 
and add the results together, putting dy instead of 


a H aa 2 aad H de, 


and du ibaa of 


we shall have 


& +o Si mda Se dy +S de = du = 0. 


A similar result may be deduced from the equation v=0; and 
it follows, that in differentiating the equations w = 0 and v = 0, 
with respect to all the variables, s, t, «, y, and z, and in substituting 
instead of dy and dz, the expressions for those differentials, con- 
sidered as functions of three variables (146), we must make the 
coefficient of each independent variable separately equal to zero. 

By considering the differential coefficients themselves as new 
_ functions of the independent variables, we shall have no difficulty 
in investigating the differentials of higher orders ; we shall therefore 
conclude what remains concerning the formation of differential equa- 
tions by some remarks on the elimination of constant quantities 
and of functions. 

150. The equation « = 0, between x, y, and z, having two dif- 
ferentials of the first order, equations (1) and (2), art. (155) ; it is 
evident, that we can eliminate two constant quantities between 
these three equations, and the result will express the relation be- 


: dz a 
tween the variables, x, y, z, and the coefficients ay OF eee Pay 


dently of the particular values of the quantities sh ali 

If to the above equations we join the three (3), (4), (5), of the 
second order, we shall have six equations, from which we may 
eliminate five quantities, and so on. 

151. This leads toa very important remark, that we may eli- 
minate from an equation of three or more variables, functions, 
whose form is absolutely unknown. 

Take, for example, the equation 


wz s2f (ax-+by), 


in which the -characteristic f denotes a function, whose form is 
indeterminate. From this may be deduced an equation between 
» 


* 


net 


196 THE DIFFERENTIAL CALCULUS, | 


and id which is independent of that function, and which is 
OL y ; an | 
equally adapted to z = aa-+by, toz = .f (ax-+by), to z = sin 


a i (ax+-by), and in general to all functions of the quantity ax-+-by. 


op Making ax-+-by =1, then the given equation becomes z = F's 
i d (t 5 
and consequently, we have dz = f'(t) dt, denoting 7 by f'(Os 
but 
. dz dz 
dz = "hs ae dy; 


dt dt 


whence , 
dz dt dz AN sdb 
= Fh) ar = FM) ; 


dt 
ie and 


JF’, we shall have 


si 


t , ih, Fi 
putting for dy’ their values, a and 6, and then eliminating 


This equation expresses a character by which we may dis- 
tinguish whether any proposed quantity is a function of ax-}-by or 
mot; for from its formation it ought to become identical, whenever 


: “taal d. 
» we substitute in it instead of ‘ and 4 their values deduced from 


any function of ax -+~ by. Let us suppose that we are unacquainted 
with the origin of the polynomial ax?+-2a bry+-624? : by equating 
it to z, and differentiating, we shall find wet 


dz 


| dz 
=z Qa? _= . 
Te 2a7n-+ aby, dy 2abr+-2b7y ; 


these values substituted in the equation 


Cicil de . 
Piola | . | 

render it identical: from this we conclude, that the polynomial 
represented by z, is a function of ax+-by, whichis otherwise ap- 
parent, since / | ! 
a’x?-+-2abasy--b?y? = (ax-tby). . « 


aa 


¥ 


‘oe Oe | . 


THE “DIFFERENTIAL CALCULUS. 157 


In general, when we have any equation u = 0, between a, y, z, 
and any indeterminate function represented by f(t), and in which 
only tis given in. terms of x, y, and 2, we may always elteHngts 
f(#) and f(t), by means of the equations (1) and (2), and u == 

When we proceed to the second order, the number of edubiions 
becoming greater, it is impossible, in many cases, to eliminate two 
unknown'functions ; but we shall not enter into these details, nor 
into those which relate to equations containing more than three 
variables, 


~ 


‘SECTION XIV. 
Maxima and Mimma of Functions of two Variables. 


152. If, in a function of two variables, we consider one as 
constant, and if we give to the other an infinite number of different ” 
values, for each of these values the given function will have one 
or more values, and amongst these some may be maxima, and 
others minima, which may be determined by making the diffe- 
rential coefficient of the function taken relative to the quantity 
which is supposed to change, equal to zero. 

Thus if we suppose w to be a function of x and y, and if we 
suppose y to be constant, and make © 


au 
& 


dense 
we shall obtain those values of x, which make wu the greatest or the 
least, when y retains the same value. 

The result which we thus obtain is still indefinite, because it 
May vary on account of the changes which the other variable y 
undergoes ; and it consequently only determines the relative 
maxima and minima, arnongst which there exists a certain number 
which exceed, or which are less than the others, and which cor- 
respond to determinate values of y. These latter, which are com- 
pletely determimed, are the absolute maxima and minima of the 
given function ; they may readily be discovered, by eliminating x 


158 THE DIFFERENTIAL CALCULUS. 


from the function u, by mage 0 of the equation S* = = 0, which will 
render ua function of y only, and denoting the result by v, we 
must then make a == 0, which equation will determine the values 
of y. 
We may obtain an equation equivalent to 7 = 0, without eli- 
minating « ; for this purpose we must hed that the equation 
du 


We =0, which arises from the condition of the maximum or mi- 
2 


nimum, relative to x, establishes a relation between the variables « 
and y; so that the former of these may be considered as a func- 
tion of the latter. By differentiating upon this supposition, we 


have 


du dx . du 
et ieee {) 
dx dik ay ‘ ‘« 


which result is reduced to - = 0, because from the condition re- 


du 
lative to x, we have already found * = here 0. 


Those values of z and y, which correspond to the absolute 
maxima and minima of the function uw, may therefore be determin- 
ed by means of the equations 


By extending these considerations to functions of any number 
of variables, we shall find, that in order to discover the absolute 
maxima and minima of these functions, we must make their diffe- 
rential coefficients of the first order, taken with respect to each of 
the variables on which they depend, separately equal to zero. 

153. The characters which distinguish the mazima from the 
minima, in functions of several variables, may be deduced from 
principles analogous to those which have been employed with 
respect to functions of one variable (76); but the appiication is 
more complicated : we shall therefore confine ourselves to that 
which concerns functions of two variables. 

Let u be a function of x and y, and if w =f (xtz, yok) 's 
such a system of values be supposed to be assigned to the slit 
©, y; as renders the sign of #«—w independent of the signs of the 


THE DIFFERENTIAL CALCULUS. 159 


quantities 7, k, these quantities having any system of finite values, 
however small, and such, that the quantity w--w' will preserve the 
same sign for all systems of values oft and k between the assumed 
system andi = 0,k =0, the value of u, which corresponds to 
the system of values of the variables thus assumed, is a maximum 
if the sign of u—w’ be positive, and a minimum of its sign be ne- 
gative. In order that the sign of w—w’, in the development 


2 
: _du ,_. du ix d?u 12. diu . 
Pe Ya (a aglll >) ollbadphallg Se Medi 
— ; doa dy a 2 tad 
d?u ke 
SMES ba Be, ee ae oe Saree 


_ may be independent of 7 and k, it is necessary that such values, 
be assigned to the variables as will render 


du du 
= 0, —_—_ == 0. Ce ° ° e ° e e ; 2 ° 
dx "dy (2) 
These equations will in general give determinate values of a and 
y- In order, however, to find whether this system of values of 
the variables gives a maximum or minimum value of the function, 
it will be necessary to substitute them in the differential coefficients 
of the second order, and to determine whether the sign of the 
quantity 
d2u 7% du d2u 
ihe —— -t——_tk+_, B . w es , 3 
” dz 2 dxdy. ap (3). 


is independent of the signs of ¢ and k. For this purpose, let 
k = mi, and the preceding formula becomes 


F GU at BAC iA! abt d7u 
; dy? = O dedy dx’ 


if the signs of this be independent of m, the values of m, which 
render it = 0, must beimagimary. ‘This gives the condition 


dud?u ¢d?u 
dx? uy? — (ca 


VO cs A PMO, cd ll 


2 


That this condition may be fulfilled, it is necessary that re and 
; x 


du 
dy should have the same sign. 


d? 
{f 7 = 0, the quantity (3) becomes — sy and the sign of this 


160 THE DIFFERENTIAL CALCULUS. 


quantity; must therefore be the sign of (8), since it ann retains 
the same sign. Hence it follows that 

ist. If ne system of values of «and y, determined by (2), give 
dtu 
dx 2 
maximum oF mifimum. 


and «—~ different signs, ‘the function has no corresponding 


2 


a If any such system of valhes of « and y give * and 


af —— the same” a ih and yet render 


d?u'd2u d?u 
dx2 dy? \dxdy 


yr< 0 or = 0, 


the function has no corresponding maximum or minimum. 
i git d? 

3dly. If sucha system of values of a, y, give os and i 
a negative sign, and also fulfil the condition (4), the corresponding 
value of the function is a maximum. 
d2u dtu 
dx*? dy?’ 
fulfil the condition (4), the corresponding ‘Mlas of the function is a 
minimum. 

154. It may happen that the system of values of 2, 3 cD deter- 
mined by (2), also fulfil the conditions . 1“ 


? 


Athly. if ‘such a system render — both positive, and also 


du d2u ad?u 
de ype 7 dady 


in this case it will be necessary to substitute them in the partial 
differential coefficients of the third order. 

If they do not render these = 0, the function admits of no 
corresponding maximum or minimum ; but if they do, it is neces- 
sary to examine the effect of the same substitution on the differ- 
ential coefficients of the fourth order. The terms of the develop- 
ment involving 2, k, in four dimensions, being treated as those in- 
volving two, and the conditions of imaginary roots determined, Si- 
milar conclusions follow, and so the investigation may be conti- 
nued as in functions of a single variable. 

155. Similar reasoning may easily be applied to functions of 
any number of variables. If u = f(x, y, z), the conditions which 
determine the system of values of the variables whigh may give a 
maximum or minimum, are : 


YHE DIFFERENTIAL a . 161 


du = 0, = 0, — ‘oo . +i 9 Lake . (5). 
dx “ dz | ‘ 
But to determine if any and what system of values of the variables 
derived from these equations must give a maximum or minimum, 
it will be necessary to examine their effects upon the successive 
partial differential coefficients. It frequently happens that some 
one or more of the equations (5) can be inferred from the others. 
In this case the number of independent equations being less than 
the number of quantities to be determined, it follows that there 
are an infinite number of systems of values which may all determine 
maxima and minima. 
In this case, if the question be geometrical, the solution is a 
locus. 
156. We shall give some examples of the investigation of 
maxima and minima of functions of two variables. 
Ex. 1. To dividea quantity a into three parts x, y, and a— 
x—y, such that the product 


f 
U = Ir™y"(Aa—x—Yy)? 


is @ maximum or minimum. 
The differential coefficients of the first order are 
du 


a = gm—ly Ml pma— HEI om my — pas ‘ 


ale ; Ps. he , 
—_ = ae —T = p* igag —NEL—NYy—pys. 
ia hs See ny—pyt 

The factors ma —mx ~my — px, and na—-nxz —ny —py, bemg put 
equal to zero, and solved, give 


ma na ated? a 


} m-+n-tp’” men tp’ iin dicts m-+-n-p- 


In order to discover whether these correspond to a maximum’ or 
a minimum,-we must substitute them in the general expressions 
for 
d?u du d?u 
dat” dy®? dady’ 


and if m--n-+p be put = % we shall find 
gr du may"—1 ynay” pay e—' 
eat) GY GY 


162 


Y @ 


ey Ge 
Pu | mana m—t m1 o—1 
dady 7 me) (~ a) Le 


The quantities ins a wil are both negative, and fulfil the con- 
dy? 


dz?’ 
dition (4), and therefore the eptrespendine value of the function is 
a maximum. 

We shall not pursue here: the investigation of the consequences 
of the other factors of the above equations being. = 0, as the 
student can readily do it himself. 

Ex. 2. To find the greatest triangle which can wee tnicladled within 
a given perimeter. 

Let the perimeter be 2a, the sides x, y, and ‘lie x—y, and he 
area u. By.a well known principle, 


w= /fa(a—z) (ay) ies ; 


assuming the logarithms, 


Zlu=la+l(a—x)+l(a —y)+l(a-+y —a). "2 te 
Differentiating for « and y, we find | 
at a—c aty—a’ 
% 
_ du he ___ 2a—2a—y 
eiide pike (a— —«)(a-Fy - a) a\? 
du ; 2a—2y—ax 
= = uu. -——— 
dy (@—y)(@-+y—a) © 
The conditions under which these = 0 are 
24—22—y = 
2a—2y—x = 0, 
Hence 
mr eB 
= 3” 
2, 
y ain 3 "3 ‘ 
9 
20 wen Qoen Y S 3 Gs 


THE DIFFERENTIAL CALCULUS, 163 


Hence the triangle is equilateral. _ It is evident from the nature of 
the question, that this result determines a maximum. However, 
this may be proved by examining the partial differential coefficients 
of the second order by the criterion which has already Shae es- 
tablished. ‘ 
Ex. 3. Let u = y*— 8y?-+- 18y? — 8y-b a? 32732. 
ine = 0, y = 2, 2-+4/3, 2-4/3. 
ee = 0,2 = 14,72, 1-2. 


Tf y = 2, and z = 1—,/2, we have 


d3u af 
oe Per 
i ba d24 r { 
dydx 
du 
dz3 pees a } 
: d?u, 
In this case a ee at ~ are both negative, and 
y 


45 d? shiny d?u\? 
dy? “dx? > (cas) ' 
i v2 U = 3+44,/2, a maximum. 
If y = 2/3, or Q--,/3, ands = 1-+,/2, we find 
. d2u d2u du 
—_—— = PO EY | he ae. Os 
dy? ams dydz Os aga B/ 


d?u 
we have therefore and = ——~ both positive, and 


d?u 
dy? 
d?u d2% d?u\? 
dy? da? (2) 
consequently 


“= ~6—4,/2, a minimum. 


, | pA au 
ffy= hg or 2— 4/3, and z = 1—4/2, we find dy ea 24, 


d?4, ae alto du t 


164 THE DIFFERENTIAL CALCULUS. 


different sigs, and the function wis neither a morimum nora — 
mINtnum. mt bbs 
Exy 4. Let uw == 2? 4-xy-fy?— an by: 
9a —b oe ESSE pene 
g ths = —, and y = ‘gra we have 


d2u d?2 4% d? u 
—. Sx De ieee] pote at 8, and 
dy" 4 dydx 1, dx? ” 
d2y,-d2u4 a2.u 
dy?” da? dz? > (Ge) 


~G2—p2 
Saar dine tk a?—b 


—, a minimum. 


Hx. 5. Letu = a} y—3ary: ys 
If x = 0, and y = 0, w is neither a maximum nor a minimum. — 
If x =a, and y = a, we have 
ay dy du 
== Ga) s)GeaBq, == <= 6a, and 
dy? hte dydz "dx? 4 
du ee d?u 7 
3 dy? “dx? (sas) ; 
consequently « =—a®is a minimum when ais a positive, and a 
maximum when @ is a negative quantity. 
Ex. 6. Let u= 2ry+2x2z+2yz: which is subject to the 
equation of condition . " . 
xyz = a%, oe 
Eliminating z by means of this equation, we find 


& 3 
= Bay poet : 
from which we readily find, making 
du du 

a 0, and --- = 0, 

x“ = y =a, and therefore z= a; and u = 6a’, a minimum. 

‘This is'the solution of the following problem: Amongst all pa- 
vallelo pipedons of given volume, to find that which has the least sur- 


fee. 


( \ ft 


we 4 


oy 


THE DIFFERENTIAL CALCULUS, 165 


SECTION XV. 
Transcendental Curves. 


157. Those curves whose equations cannot be expressed by a 
finite number of algebraical terms, are, in general, called tran- 
scendental curves. Before we proceed to the discussion of these 
curves, it may not be improper to make a few observations res- 
pecting the transformation of co-ordinates. Let us consider a 
curve BDC (Fig. 29) in which we have determined the position of 
any point MM, by means of the rectangular co-ordinates AP =a 


& 


_and PM=y; this point may be equally determined, if the angle 


MAC and the radius vector .1.M, be given ; but as angles are com- 
monly measured by the arcs subtending them, we can replace the 
angle MAC by the are mo, described with a radius taken for unity ; 
thus denoting the arc mo byt, and the radius vector 4M by wu, 
we could substitute the system of polar co-ordinates ¢ and u in that 
of the rectangular co-ordinates 4P = xz and PM = y. 

158, Let us represent by f(z, y) = 0; the equation of the curve, 
in which we would wish to transform the rectangular co-ordinates 
AP =x and PM= jy into the polar co-ordinates om =. ¢ and 
AM = u, and let us find the relations which exist among these co- 
ordinates ; we shall have evidently 


AP = AM cos MAP, PM = AMsin MAP, 
or | 
eu COS ty U SINE. eis ee we, (A) 
It would be sufficient to substitute these values in the equation 
represented by f(x, y) = 0, in order to obtain that which should 
be referred to polar co-ordinates. 
159. The equation referred to the polar co-ordinates being repre- 
sented by f(t, uw) = 0, we see at first (Fig. 29), that we can re- 
place w by its value derived from the equation 


AM? = AP?+-PM?’, 
or 
42 — z?-+-y?, e ° ° e Py ry e (2). 
& 


Se 
i 


i66 THE DIFFERENTIAL CALCULUS. 


With regard to t, the equations (1), divided one by the other, 
will give 


hence | — 


. This value of ¢ and that of w. derived from equation (2), being 
substituted in the equation represented by /(i, u) = 0, we obtain 


f} arc (ta = 2), J (2?-+y?) =0... * + (8). ) 


We arrive thus at an equation ‘in terms of the rectangular co- 
ordinates x and y, and affected with a transcendental quantity. . 
160. We may also obtain between « andy, an equation which — 


would not contain the tréttcenflenial arc (tan = Es : but which 


would comprehend differentials; by differentiating the equation which 
is represented by the formula (3), or, we may employ other means 
to arrive at the same end: let us always represent by f(t, w)=0, the 
equation which is required to be transformed into a function of the 


rectangular co-ordinates x andy; we have already seen that the . 


fag ah 
value of ¢ contains the transcendental arc ( tan = ‘), and that, at 


the same time, the value of u is expressed without any transcen- 
dental. i 
It would be therefore necessary to eliminate ¢ between f(t, u)=0, 
and the differential of this equation, which we shall represent by 
f(t, wit, iu)=0, in order to find an equation without a transcen- 
dental; in fact, by this means, we shail introduce into the result of 
the elimination, the differentials /¢ and du ; but these differentials 
may be expressed in functions of the variables a, y, dx and dy. 
In fact the equations (!), give 


cos t ==, sint = oo ee ye ae - (4); 


w 


dividing the second of these equations by the first, we obtain 


sin t . 
»—— ortant = a , 
cos ¢t R ‘ 


differentiating it becomes 


oe 


er 


ooo 
° 
THE DIFFERENTIAL CALGULUS. 167 
dt £ ady—yda 


cos? ¢t * 2 
Ji 


Replacing ae by its value derived from the first of the equa- 
fions (4) and suppressing the common divisor 2?, we shall find 
o® wdt = xdy—ydz, 


and consequently — 


at = EES 105 


meat 6 Bre MOM LR Gane 


and by putting for w its value, this equation becomes 
andy —ydx 
roe 
The differential of the other variable is more easily found ; since 
equation (2) gives : 


dt = 


“= / (a?+-y?), . 
this equation being differentiated, we shall have 


yOaty) » 4 
by means of these values of dt, of du and u, we shall change the 
equation obtained by the Slimination of ¢, into another which shall 
contain only the variables x, y, dx and dy, and which shall be con- 
sequently referred to rectangular co-ordinates. 

161. We have seen (10) that the differential of an arcs, re- 
ferred to rectangular co-ordinates, has for its expression 


ds = (du2+dy). 2 2... (6): 


in order to determine the differential of the same arc, when the 
co-ordinates are polar, we will substitute in equation (6) the values 
of dx and dy, derived from the equations 


z =u cost, andy = u sin t, 
and we shall find, by differentiating these equations, 
dx =—w sin t dt--cost du, 
dy = ucost dt+sin t du; 


% 


squaring these equations, and observing that — ‘ 1 oe 
& 


* . 
, 


2] 


#, 
* 


i6s THE DIFFERENTIAL CALCULUS.. 


‘gin? trb-eostt = 1, 
we shall obtain | 
= Yoedi du’ ) 


Such is the differential o hs arc in a function of polar corordi- 
nates. 

- 162. In order to determine the analytical expression of the sub- 
tangent in polar curves, let 4.M and AM (Fig. 30) be two radii 
vectors, and from the point JM draw the perpendicular .“P upon 
the radius vector 4M’, and, to this perpendicular, draw the parallel 
AT’; the similar triangles 47.M, PMM’, give 7 


PM PM aM oar; 
whence we derive 
| es. sr XPM _ _AM XPM 

eo eG V/(MM?_ PM * 


in taking the limit, 4M’ is equal to 4M = u, PM = the are JUN, 
the chord 4M’ = the arc MM’, and AT becomes the subtangent : 
but the differential of the arc MM’, = 4/ (udt?-+-du*), (160), and 
MN = udt ; for the sectors ARR’ and AMN give - 


ae AR: RR +: AM: MN, 
or - R Bes : Uw: MN ; ec 


7. MN=uRR,a pitioaitie wich Mine the limit, eaptile udi. 

Therefore, by putting these values of MN and MM’ in that of 
AT, and observing that 1M pasta = a and PM = MN, we 
shall find ; 


such is the expression of the subtangent.. 

163. In order to determine the subnormal, let us observe that 
the normal SJM being perpendicular to the tangent, the ordinate 
AM (Fig. 31) must be a mean proportional between the tangent 
and the subnormal ; consequently we have 


(hyd AT: AM:: AM: subnormal, Sala mn 
or | 


——: &::u: subnormal : 


y yet 


~ 


» 
ey 
* * % ‘ , 
rs he a 4 
YHE DIFFERENTIAL CALCULUS. 169 
if subnormal = —- 
dt 


4 


ae: 
“With regard to the normal. and the tangent the right angled 
triangles MAS, MAT, give a. 


“MS = SMP LAS, MT = = ata pats} 


substituting in these equations the values of MA, AS, and AT, we 
shall find 


normal = a/ co , tangent = wu a/ 5 + te bet Ci 


164. The analytical expression of a sector in fier curves, is, 
readily found from the area of the triangle AMM; for, in taking the 
limit, the area of the triangle AM'M becomes that of an elementary 
sector, the perpendicular PM may be replaced by the are MJY, which 
~ we have found equal to udt, and de coincides with 4M = wu ; 
therefore, since 2 he 

aoa! 


AM'XP 3 
et area f wl MM == PEL art 3) 


we shall find, is Pa 


u'di 
area me an elementary sector = 


We may also. express the area of an ele Piantnit sector in a 
function of the rectangular co-ordinates ; for, by putting in this 
equation the values of u and dt, given by the equations (2) and 


(5), it becomes - | ; . 
he ndy —ydx 
area of an elementary sector = = TE. 
al ‘e 


me 
ape 


165. Let us now consider the radius of curvature : and here 
we must observe that the formula (123) may be written thus: | 


In order to have this value of ¥ saomccay in a function of polar 
co-ordinates, it is necessary to eliminate the differential coefficients 
which enter into the expression by means of the equations. 


HT Z We 
& ’ 


co i es 

170 THE DIFFERENTIAL CALCULWUS.. 

we ai dx = 24 cos t—udt sin ¢, 

i dy = = udt cos ttodu sin t; 

let the second equation fe divided by This first, and, putting 


~- 


‘ m = udt cos t-t-du sin t 1 
- . 4 (8 )s 
) Pee du cos t—udt sin t 
‘a " : : ‘p 
we shall have 
hina dy m oo * i 
y 4 in. = beeper } ry aa ‘ot e o a ° (9), 
eck dx % aa SEN “Her 
or ) 
ee dye ae 
| Re ' dx? n? pt 


By means of this equation we find for the numerator. of the 
, ‘value of y, 


pee 


f 2 
coe 
‘ 


Poa. aw 


(14 ae = n° oa _ (n?--m)2 
, a ee 


Differentiating equation (9), we shall find 
» dy __ ndm—mdn, 


= 
rs a 


dx rr, Mes 


dividing the first term of this equation by oe and the second by 


2, which is equivalent to it, we shall have 
re 


@ da? = pe “Ge ap » orl). @ le (11). 


' By means of the values given by equations ie”) and (11), the 
ii equation (7) becomes 

2 2 ; dt . ‘ ve 
(tema) ee es (39). 


~ ndm—mdn 


There now remains only the transformation of this equation 
into a function of ¢ and uw For this purpose, we shall at first 
_ determine the value of n?-++-m?, by means of the equations (8) ; 
that is to say, by squaring the equations (8), adding the results 
together, and observing that sin’ ¢ -+ cos? ¢ =_1, we shall find 


} lig @ dufedAsse ant (i dese 088). - 


? 


{THE DIFFERENTIAL CALCULUS. V7} 


With regard to the denominator of equation (12), let us differ- 
entiate successively equations (8) and making dt constant ; and 
multiplying the results by mand m pespecinelys we shall find 


rt 


_ndm = nd*u sin tpn cos t—nudé sin ¢, 
mdn = mdu cos vis 2mdudt sin t—mudé? cos t * 
the second of these equations being subtracted from the first, gives 
ndm—mdn = d*u, (n sin t—m cos t) | 
, PSs 2dudt (ncost+msint) >. wen (14) ; 
- udé? ip sint—m cost) |. D 


multiplying the secon of equations (8) by sin ¢ and the first By. 
cos‘ and subtracting the results from one another, and reducing 
by means of the relation sin ” eacos" ¢ = 1, we shall obtain 


nsin tm cost == —udt. Ws 


Multiplying the second by cos ¢ and the first by sin ¢, and pro- | 
ceeding in in a aemalar manner, we shall find 


n cos t+-m sin t = du. 
Substituting these values in equation (14), and it becomes 
ndm—mdn = —ud*udt-+-2du*dt--wde . . «. (15). 


By means of equations (13) and (15), equation (12) willbe 
reduced to 


(du uP dt?)? 2 


 Qdut Bdul dé—ud'udi-pued® © © °° (16) ; 


which is the radius of curvature in terms of polar co-ordinates. 
166. Weshall now proceed to the investigation of some of the | 

most remarkable among the transcendental curves. The loga- 

rithmic curve will first engage our tte, in which the ordinates 

_ are the logarithms of the abscissz. 

The equation of this curve is therefore 


y = lex; 
whence we derive by differentiation 
dy 1 ; 
de oe 


* * 4 


172 THE DIFFERENTIAL CALCULUS. 


we see by this, that the tangent of the curve is sean to 
the, line; of the abscisse when « = 9; and that it only becomes 
parallel to it, when z is infinite. The general expression (96) gives 
the subtangent == xy; but the elimination of y introduces the 
logarithm of x; so that this expression is. transcendental. If, 
however, we find the value of the subtangent upon the line of the 
Asana sa we get | 


subtangent = =~ bacrispt S45 
dx 


Now the first member of this equation expresses the subtangent 
of the curve, therefore this subtangent is constant and equal 
to unity for all points of the curve. We should find too, that the 
tangent, the normal, and the subnormal, taken with respect to the 
‘line of the abscisse, are transcendents, the ordinate y entering 
into their expression ; but that they become algebraic, when con- 
sidered with respect to the axis of Ys 


ty We e proceed to oun? the radius of curvature. Now, we 
have | 


= sachs, a+ d2y 1 
i MA iy 
Ta a?” dx? ze 
whence (165) 
jilies 
ppt a ? i 
& - 
y-B = V1, t— 4 = haa / ’ 


Or, if ¢ = the modulus in any system of logarithms, we. shall 
have | 


(x? fe?)? 
a 5 AM 
fm Bao ag Lm OS aoe eo 


Logarithmic curves differ from each other, on account of the mo- . 


dulus relative to the system of logarithms, which they Prepreseag 
The equation of logarithms, 


na 
being differentiated, gives | } 
dy ‘i | 
ian @ la ; 


THE DIFFERENTIAL CALCULUS. 173 

whence we deriye iad sl is she 
an * ¥ my a i Aid 
atdx ude My Ha | ¢ 


dy iz eats dy la” ; 7 


Now, the first machider of this equation expresses the meitenaent 
of the curve; therefore this. pl ae is always constant and 
equal to the modulus. 

167. The spirals compose a class of transcendental curves, 
remarkable from their form and their properties*, The loga- 
rithmic spiral is a curve in which the angle AMT (Fig. 31), formed 
by the radius vector 4.M, with the tangent MT to the curve, is con- 


stant. Thus, denoting the dah ule tangent of the angle 
AMT ie a, we shall have 


ee ce talgs tc: th AMEE o's 


q 


nee the right tak a TMA, gives 


| 1: tan AMT: : > AM: AT; ial 
| AT 
. tan AMT = ar 


Putting u for the mas vector AM, and — Th ft +161) for the suite 
tangent AT, we shall have 


tan AMT, or @ “Ba 
du 


ia 


whence we derive 


and 


pus wif We Was, ee ken NT 


. . i . 
differentiating equation (1), dt Fis constant, we shall find 


3 
uy = oe eee. SAO 0 CRD FO 


* The reader is referred to the Appendix at the end of this work. 


+ 


* 


+. % bes. PS, 
ee. € 
174 THE DIFFERENTIAL‘CALCULUS. 
Substituting these values (2) and (3), in that of y; equation (16) 
art. Mp) we shall obtain % ie. 
i a: ; Ab, rf i Mui ar 
BP . 2 2 a het) Sah Ny ts 
' oa u? . Uys . e y2- 
SE — Sangam (ei) ; =Y in wu? é 
a 


“Again, if i ig het expression ‘of the normal (163), we substitute 


the value of - a we shall find ¥ ie ie, ie ak, Sabie aS 


~~ pormal = v 8 ie Hie pant th : 
which proves that the normal, in this curve, is equal to its radius of 
curvature, and both being in the same direction, it alse follows that 
these lines must coincide. a 


168. Let us now consider the spirals comprised in He equation 


ts at”, | 
having all possible values given to it. If n= —1,we have ui = a, 
1 
an equation which belongs tp the hypectoins spiral ; ; if c= Dn” we 
obtain the spiral of Conon or Archimedes ; ; and if 1 n= 2, the equa- 
tion u = at’ would be that of the parabolic spiral. iin. 
The above equation may be put under the form 
OD sche 
, uv =a t, ™ ~ 
which gives by differentiation 
1 ft pt 
Wad na 
—-u db=—a dt; 
% 
whence we derive % 
wdt a : 
tangent = ——-—~ ¢ nti, , 
bite | du on +) teenie . , 


° ; aah j 
In the case of the spiral of Conon, we have n = 1, anda = ot 


and consequently, “the 


2 


Pu 
subtangent = on" 


a 


© 


DHE DIFFERENTIAL CALCULUS. Vs 


From this! expression, we see that when ¢ = 22, or after one revo- 
lution of the radius vector, the subtangent is equal in length to the 
_ circumference of the circle, at the end of tivo revolutions the sub- 
tangent | will be equal to four times that quantity ; and so on, as Ar- 
chimedes remarked. 


When n =—1, which i is the “case of the byegeoue pire’, we 
have se . “ee 


hs i 


i =—d, 


that is to say, the. subtangent of this curve is constant. ere 

It is not necessary to consider particularly the expressions of the 
normal and subnormal, because they may readily be obtained when 
the subtangent is known. 

169. The Cycloid, or the curve described By a point in the 
circumference of a circle, while the circle itself rolls upon a right 
line given in position, is “another transcendental curve ; the rela- 
tion between its ordinates and abscissew depends on the arcs of 
the generating circle, and mayne expressed in the following man- 
ner. © a ee 

The origin of the motion of the uicle Shing arbitrary, we will 
assume. the point A for it (Fig. 32), where the deseribing point 
was situated in the right line 1B, which the generating circle 
Q.VG rolls along. Since the circle in rolling applies every point 
in its circumferenée to the line AB, it is plain that in any situation, 
as QMG, the distance 1Q is equal to the are MQ, contained be- 
tween the point M, which at first touched the line 4B in A, and the 
point Q, which is in contact with it in its present position. 

If on AB, we erect, at the point Q, the perpendicular QO, pass- 
ing through the centre of the generating circle, and draw MN 
parallel to 4B: JMN will be the : sine of the are MQ, and NVQ the 
versed sine. 


Let QO = 6, P= PM = QN=y, 


and we shall have 


MN = ¥ (2ay—y"), 2 = AQ—PQ = are MQ—MN, » 


- tS ~ 
*: _ & = are {vers = y5 — —JS(2ay—y) ss ee (1). 


which is the primitive equation of the cycloid. The are MQ 
(whose versed sine is y), has also MN, or f (2ay—y?) for its 
sine ; the foregoing equation may, therefore, be written thus ; 


& 


Be ° 


176 : THE DIFFERENTIAL CALCULUS. 


a == are {sin = /(2ay— yt EBay sey din Roars (2%. 


Differdutiating this equation, the circular are will disappear ; for by 
the formula (105), in which a represents the fei and 2 a ae 


aN we substitute v (2ay —y?) for xs we shall ign Poe Whats 
| ye ere as . om % 
a arc oad | _ ail. ad. «/! Day a ee YER. ys 0 
a O f(ai— day ry) 


> ws a—y 4 ad } : ty 2 hts th , 
ae he ; ' : A ‘ 


fy i aes by substitution : fe ¥, ss 
d. are. MQ = 2° (om )dy oB par ail oat 
De ag ce F) a 4 ‘eae 
Whence also, | Git, Mtge Wag 
oe atly © ; vada Phy " 
. | Vea uy) v Oa hcg ly Se, 
and. consequently, ok s tg. ‘ a 3 
= Fey yy) , 


whish’i is the differential equation of the ig dioidit , 
We can ‘readily obtain expressions for the subtangent, the tan- 


‘gent, the subnormal, and the no “aan in the se hee find by 
the anne formule of art. nf: 


. 2a Je, 

yp PTs MT = ae (ay 
| V/ Feias | Vv v (2ay—y") oe 
PN = / (Qay—y? ), MN = J (2ay) Pye 


We may construct these values in a very simple manner, for it is 
easy to observe that PM or y being considered as the abscissa Qn 
in the generating circle QMG, the value given above for PN, is 
precisely that of the ordinate MN of this circle, and consequently 
the normal coincides with the chord of the are MQ, as a be 


* The sine wtike nin to ‘radius a, that of the tables, having =a for its ra- 


dius, should pe ees =¥) a. If we wish to introduce this : sine, for the purpose of ~ 

calculating the length of the arc me, the expression must be written / 

oes a arc (sin =  /(aay—y2)- J (Qay—y2 y. " % : 
” at ly ; % ' a e “aye 


aa bats 


# 


me CHE DIFFERENTIAL CALCULUS. 177 
also seen from the "6 expression for MN. It follows from hence, 
that the chord JMG produced is a tangent. If we conceive the 
circle QMG to slide upon the point Q, so as to arrive at any other 
position gmg, the lines mq and mg will continue, notwithstanding 
this change, parallel to MQ, MG. It is, therefore, sufficient for 
constructing the tangent andnormal at any point WV, to refer this point 
to the fixed circle gmg, which may be done by drawing Mm parallel 
to .4B, and then to did MT parallel to mg, and MQ parallel to mq. 

170. Let us next consider. the radius of curvature of the cy- 


_ cloid. From the differential equation of thé curve we have 


dy of (2ay ~ 4) 
‘dx rae Fadia . ° . e e cy e (3), 


d | 
in order to obtain the value ofa da® ¥ Tet us make = =p, and we 


ie 
Hon 


wit 


” . y 
ag *N, 
by aitreaiaon, we shall find 


ee dp _ a 
Gy, ollie I Oo oe Ml 


shall hav 6 


Multiplying this by equation (3), and we shall obtain 


substituting now ‘this Talus i that of rs in the expression for th e 


ay 
radius of curvature, (128), 
° ‘ty 
ii a 2 3. %, 
-. +5) Ea 
S de2 ; 
bY a oe a 
“4 ' ae Le) 
nae shali obtain 
. e 2 ee 
: Oe eee oe 
| p= = Ss 9247Q2 - 
x ° yea 5 
6 amit Fe, 
a y 
a8 2A 
* # 


178 THE DIFFERENTIAL CALCULUS. 


oe 
2 


or y=2. 22a y? = 2,/(2ay). 


This result shows that the radius of curvature MM’ is double the 
normal MQ : and that it can never exceed, therefore, twice the dia- 
meter.of the generating circle, which diameter is at once the ordi- 
nate and the normal to the cycloid at the point J, at which the point 
of contact Q, has traversed one half the circumference. 

171. The expressions for xa and y—£, give 


 ¥-B = 2, Cm = —DQaf(Laymy”) 5 
whence we conclude, that 
y=—f, a = a—2,/(2ap—p*). 


If we substitute these values in the primitive equation Q) of the 
cycloid, and make the necessary reduction, we find 


# = are {=—p + 4/( -2ap—8%), 
a result which has a great analogy with that equation. The radical 
4/ (—2aB — 8") becomes similar to ,/(2ay—y?), when we make 
6 =-—2a+8', which comes to the same as taking, instead of the 
ordinate EM’, which is always negative, the ordinate PV’, refer- 
red to an axis .1'B’, situated below AB, at a distance AJ = 
By this transformation it becomes 


@ = are {vers = 2a—p’}-+4/ (2ap’—p") : 


but we must observe, that the two arcs, whose versed sines are to- 
gether equal to the diameter, are supplements of each other ; and 
denoting the semicircumference by 7, we may, therefore, write the 
above thus : 


ae = ©—are jvers = B+ (2ag' 9), 


Taking then « = 7 =a! ; that is, substituting for the abscissa 1E, 
another abscissa ’P’ = AI—AE, we shall find 


e’ = are } vers = 8't—4/(2ap'—f"), 


the equation of a cycloid whose origin is at the point 4’, and which 
is described upon the axis ./'B’, by the same generating circle as 
the proposed, but rolling in the picseans AB’, opposite to AB. 

The same consequence may be also obtained from the determina- 
tion of the radius of curvature. Producing GQ to meet .1'B' in Q’, 
and drawing QJM’; we shall have the triangles GMQ, Q.M’'Q’, equal 
te each other. The angle Q.M'Q is, therefore, a right angle; and if — 


e 


AE 


THE DIFFERENTIAL CALCULUS. 179 


acircle be described upon QQ’, as a diameter, it will pass through 
MM’, and will be equal to the generating circle. This being premis- 
ed, since the are J/'Q’ is the supplement of MQ, which is itself equal 
to MQ, we shall have 


are MQ’ = QMG — arc MQ 
= J—-AQ = Qil= AQ; 


which proves very clearly, that the evolute M4 isa cycloid de- 
scribed by the circle Q.V'Q,, rolling on .1'B’, from 4’ towards B’. 

172. The reader will have remarked, doubtless, from what has 
been said before, that the cycloid is rectifiable, since it is its own. 
evolute, and the expression for its radius of curvature, is algebraic ; 
and we thence deduce a curious result, that the length of the arc 
VA, or its equal 1K, which compose the half of the branch describ- 
ed by the generating circle, ts precisely that of .1'K, or double’'the 
diameter of the circle. 

The cycloid is not terminated in L, when the circle has deserib- 
ed its whole circumference on 12; for there is nothing to limit the 
extent of its motion. We ought particularly to remark, that in the 
description of curves, all the different parts which result from the 
same construction, or from the same motion, belong to the same 
curve. ‘Thus the circle QMG; by continuing to roll on the right 
line 1B, beyond the point L, describes a series of portions, similar 
to 4KL ; and we must conceive as many to have been described on 
the left side of the point .4, since the circle may nave arrived at this 
point in the course of a motion which has already continued an in- 
finite length of time. ‘The equation of the curve leads naturally to 
these remarks ; forthere is nothing to prevent our supposing the 
are QM to increase or diminish, by as many circumferences as we 
please. We see, too, that y can never surpass 2a. Hence it fol- 
lows, that the cycloid conceived as existing in its full extent, may 
be cut by the same right line in an infinite number of points. 

The differential coefficient of the second order, 


is always negative, since y? is always positive ; but when y = 0, it 
di 
becomes infinite, as well as =, when y = 0, which happens when- 
x 
ever the are MQ is either 0, or some multiple of the circumference. 
The points A, L, &c. therefore, where the different branches of the 
cycloid touch each other, are cusps of the first species, at which the 
tangent is perpendicular to the axis of the abscisse. 


+ 


% 


Ove 7 ‘ ALS a Se t ates A ae . } i a if 


: hia cackahiias aay Tor 
. oo poi quilamasy of 


omy ¥ sie « tty 


me ta ry <a 


) ; ‘o 
‘ oat NP 
ab A occu | 
k DW CPE | CPs Te OE i 
y * = ‘ < u, 
Ww af ee) re ot ohh alent 
M bf “abt hit 2 ay Ni Page Hh . ” i 


awe Gti 2 Ab ox te oheibhisoy of | 5 

aes : 4 cabatorngsat Vio Hulk og 9) ig ry ais Sera. it og ye 

- ok " ARH iat iat Shae +) ‘hes ‘fi TMS F ar le re See ; 
abel athdh dart fey Wate! ue iy wiles rhe posi ea. th or as rH 

adv otdoos’ ta Od Tae aby eh os i ei , 


x. te 
es GP 9: pabha 


alt wait} Ait 
4 i eT ett et : 1 ae i 


ee tee, 9 


~~ wUhy 


oh Siac 


tdgi~ oe ner fiod if 
qeliatey: ctu ito ts yeh Pa a hy 
J they feat a” Vi ral ite oa men * garth f SYa8 43 wey 


‘Ries iw hove writ ys Pah hy Atk aaule %, eed ; 
yy fh: Gumarert cries ¢ aed cig: bebe reef : 
ai Ae eh arbre: ahhe- 0 Wo the ee Ap everday ret 
1 ah Rea pene aig pirate on: geridor Re ; . wi | 
ye ey & re > isi geet. ae ese sie fy ies tig? a F 


y fe eh oh rm ove 
ESET ta 4ae Hit vials A > git ie 
+ ah gett Sebi deen ait yt) “ 


ae SHU i a 


wig 


eee ste? wat 


LPR ES | a Ge 


ars 


t all 


THE 


DIFFERENTIAL AND INTEGRAL 
CALCULUS. 


* 
PART THE SECOND. 


THE INTEGRAL CALCULUS, 


SECTION L 


THE INTEGRATION OF RATIONAL FUNCTIONS INVOLVING ONLY ONE 
VARIABLE. 


173. The object of the Integral Calculus is the determination 
of the primitive function or equation from whicha given differen- 
tial, or differential equation, may have been derived. 

The primitive function is in this case called the integral of the 
proposed differential, and the process by which it is determined is 
called integration. These terms, integral and integration, are 
taken from the infinitesimal calculus, and have their origin in notions 
of this science not consistent with the rigour and purity of ma- 
thematical reasoning. As, in the infancy of the science, differ- 
entials were considered as infinitely small quantities: so the ori- 
ginal functions from which these differentials were obtained, were 
taken as the sums of the infinitely minute elements ; and the process 
by which these primitive quantities were found from their differ- 
entials, was looked upon as the summation or integration of the 
small component parts, and the operation was expressed by the 
character f prefixed to the differential, thus fxd, asthe initial of the 
word sum or summation. Modern mathematicians have reduced 
the science to more rigorous principles, but they have retained its 
former phraseology and symbols. Lagrange alone had the bold- 
ness to attempt a revolution, not only in the principles, but in the 
language and algorithm, or notation of the science; but he can 
enapealy be. considered to have succeeded, at least in the latter, 


t 


182 THE INTEGRAL CALCULUS. 


since all mathematicians, almost without an exception, adhere to 
the old symbols, though some of them use the principles and rea- 
soning of Lagrange. 

174. According to the language of Lagrange, the object of the 
integral calculus is to determine the primitive from the derived 
function ; or, if applied to equations, to determine the primitive 
equation to a given derived equation. 

According to the more commonly received phraseology, this 
branch of the science consists in the determination of the function, 
of which a given function is a differential coefficient, or the equa- 
tion, which differentiated, would produce a given equation. As this 
process is exactly the reverse of that which foyms the subject of 
the differential calculus, so the rules and methods to be used in it 
must be discovered by retracing our steps in that part of the 
science. * 

175. We shall, in the first instance, confine our attention to 
those differential coefficients which are functions of a single va- 
riable, and, as in the differential calculus, we shall successively 
consider the cases where they are algebraic and transcendental 
functions ; algebraic functions being divided into, ‘st, rational and 
integral, 2d, rational and fractional, and 3d, irrational ; and, tran- 
scendental into, Ist, exponential, 2d, logarithmic, and 3d, circular. 

176. As the differential coefficient of any power whatever, is 
found by diminishing the exponent by unity and multiplying by the 
first exponent (25), so a differential, whose coefficient is a power, is 
‘ integrated by increasing the exponent by unity, and dividing by the 
increased exponent. Thus, since 


de x™t = (m--+-1) ada, : Y 
we derive 


ight. gome 1 


“m+ 


and as the constant m-}-1 has no effect upon the differentiation, this 
equation could be written thus, 


d. (= )=- Mele = 


new an the quantity which, by differentiation, has given x"dax 


= gmdz ; 3 


pet Lites ’ iil oy tee 
aalit In order to indicate this operation us put before the 
differential, the characteristic f, which, as ha is 


ed, signifies sum or integral, and we shall have. 


JHE INTEGRAL CALCULUS. 183 


a amit 
™m = 2 
fe x m--1 


Hence we conclude this general rule : in order to integrate x™dx, 
we must increase the exponent of the variable by unity, and then 
divide by the exponent thus increased and by the differential dx. 

This rule is applicable to the integration of all differentials which 
can be reduced to the ine Ax™da. 


Let, tiiariple, * eee be the differential whose integral is requir: 
am 
ed: 


ads 5d aa—sit ax Cb 
te Se Ee i ieee ees Serene 
ra —3+1 are ‘Qe 


we shall find in like manner that 


fax. a? = fa Mar = = = See 
; a! 


176. It must be observed that, as an independent constant con- 
nected by addition or subtraction. with any function, disappears by. 
differentiation, so it should reappear by integration. Thus, if f’ (a) 
be the differential coefficient of f (<), it is also the differential coef- 
ficient of f (x)-+-C, C being a quantity independent of x. It is ne- 
cessary, therefore, to add to every integral a constant, which is ge- 
nerally called the arbitrary constant, because its value cannot be de- 
rived from, and does not depend_on, the differential coefficient, but 
must, if discoverable at all, be determined by other means. 


‘The examplesin the preceeding article must, therefore, be 
‘written thus : 


forte = — fm — arth? oft — ve as = agai sort Cc: 


: dx a2 = fixide = i CG 


i77. If the value of the integral corresponding to any particulai 
value of the variable happen to be known. the value of the arbi 
trary constant, C, may be found. For, let the integral with the 
arbitrary constant be f(x)-+C, and suppose that it is known that the 
value of the integral is when the variable sis=a; ., A= 
f(a)+C: hence C = A—/f(a) and consequently, the integral is 
F(«)—-fla)+-A. If the value (a) of the variable which renders the 
integral = 0 be known, the integral is f(x)—f(a). 


184 THE INTEGRAL CALCULUS. 


For example, if we have the equation y = a2°—b, which is that 
of a parabola CBD (Fig. 33), whose origin is at 4, and that we 
derive from it dy = 2axz dx, we shall have, by integration 

y EE aE 8 SOR Sea 


This equation. may agree with an infinite number of parabolas. 
Now if we know that among all those parabolas, CBD, CBD, 
C” B' D"", &c. the curve which has for its equation 

y = az’-+C, 


should be that of a parabola which passes through the point E 
whose co-ordinates are 


b 


it is necessary that, by making « = J/ a We have y= 0, whith 


will reduce equation (1) to 
O = 5+C; 


whence we shall have 
C ae 8.! 


Substituting this value in equation (1) we shall obtain 
y = az’? —b, 


the same equation which we had before the differentiation. 

178. When the nature of the problem does not determine the 
constant, we may give it a form siimilar to that of the first term, as” 
in the following example : let 

dy = anda ; 
ys i 
av! 


ST ie Jerr a Mie, w be 1A 


it’ we represent by 6 that value of «, which makes the function 2 y 
equal to nothing, we shall have nat od? 


a 
abut ney r! 
we = 0. ne Te eho Ue (2) 3 


whenee we will derive the value of C, which heing substituted 1 in 
equation (1) shall give 


43 
fy 


THE INTRGRAL CALCULUS. 185 


a (amt. — bm'1) 
m+ 1 (3) 5 


7 


a result which only differs from the former (1) in the form which 
is given to the constant. 


179. Before we proceed, further, it may be proper to examine a 
particular case, in which the value of y, found as above, becomes 


OA. MP he 
a, it is that in which m =—1, for then we have 


eS ob ex AE ul ae 
O 0 


©; 


To find the true value of this function we must revert tothe 
rule in art. (92), and as we have seen by this rule, from Ex. 3, 


xfer : if 
page 88, that a became la—/b, we shall have, in the ex- 
ample before us, when we substitute corresponding letters, 
y= (la 1b} 5 
but when » =—1, we have dy = ax—1 dz; and therefore dy = 
gon ives 
a ® : . 
y =a (/x—I6), or y = alx-+€. 
We mighi have drawn the same conclusion from att. (31), since 
d . : A 
it there appears, that dla = = The exception to the rule in 


art. (176), presented in this case, arises from the impossibility of 
expressing the transcendental quantity Zz, in a finite number of al- 
gebraical terms. 

The whole difficulty of the integration of functions of ore va- 
riable, consists in the discovery of such transformations as are pro- 
per to reduce the proposed functions to one or more terms, involving . 
simply powers of the variable and constant quantities, to each of 
which the rule in art. (176) may | be applied. 

180. As the differential of a function, which is the algebraidal 
sum of several functions of the same variable, isthe sum of the 
differentials of these’ functions, so the integral of the sum of se- , 
veral differentials of functions of the same variable is the sum of 
the integrals of these differentials. Thus, if . 


r dy = ax™de+-badx--catdr-, Ke. 


we shall have 
2B 


186 THE INTEGRAL CALCULUS. 


Jdy =f} axmdx-+-barda--caPda-+-, &c.}, 


or 
fdy = Saxmda--fox"dx--fexPda-+-, &c. : 


and consequently 


mami! bani | caell 
powen eee -——— e e ° e e Gi 


We have added only one arbitrary constant, for it is easily seen 
that if a constant was added to each integral, they would together 
be equal to one constant eo 2g which is equal to their sum. 

For example, 


f adem bade » RY earrayy 2 is 42 yctanel c C* 
or, by effecting the operations, 
Aas bda DNs? tere 

ft} ada — pada fs —— ae te ate C. 


181. If there were given dy = (ax-+0)"dz, we might expand 
the given power, and integrate every term in the result : for exam- 
ple, in order to integrate (a-+-bx)*dx, we shall have oy 


S(a-+-ba)dz =f(a*du-+-2abrdz-+-b?2"dx) 
2 an3 
s2 ata-babat ap GC: 
But it is proper to observe that we may obtain the integral without 
effecting this development : for the rule in art. (176) extends to the 
integration of all differentials which can be reduced to the form 
Aatdx. : Such is (ax-+b)"dax ; for, if az-+b = z, we have 


z= “Mak which gives dx = ma 
a a 


by substituting in the expression for dy, we find 


M gf 
dy = a = Az™dz, 
a 
and consequently 
ite 


Anh a(m-+1) : 
Putting for z its value, we find, that when af 


THE INTEGRAL CALCULUS. is? 


t. (ax--bym't 
a(m 1). 
If we had dy = (ax"-+-b)™a"—"da, this transformation would still 


succeed ; for by making ax"-+-b = z, we have nax"—'dxz = dz, and 
therefore 


‘dy = (ax-+b)™da, y 


whence it appears that, when 


dy = (au”--b)ma"—Idz, = ana = Ce fd ibe 


This example may be also transfdemed to the firm Ax™da in ue 


following manner ; if x" = z, we have ae Sat =. dz, 


oe 2" (a-- ba") "de = : (a+ bz") "dz. 


Again, leta--bz =; .*. bdz = dy; 
hence we find 


a1 (aba) mda, = is ydy = Aymdy ; 
n—1 mn Bd 
for (a-bban de = A. wt ‘ 
The rule above alluded to, is subject to the exception f x~'dx, or 
SF ol as has already been observed ; the value of this being /x+-C, 
Under this case also come all those differentials which can be reduced 


: d 
by any transformations to the form a ; such as 


dx _ dea). 
aha ota” 5 ftir 


183. As the differential of the product of two functions of the 
same variable is the sum of the alternate products of each func- 
tion into the differential of the other, so the product of two fune- 
tions is equal to the sum of the integrals of each function into the 
differential of the other. From this principle an important method 
of integration is deduced. Let uv be two functions of x. 


= I(a--a) + C. 


te 


piles 
q 


THE INTEGRAL CALCULUS. 


1385 
Hence ’ Sa 
uD = fudv+fodu ; 
ui 2. fudy = wo—fodu. 


By this equation, the determination of one integral Jud» is made 


to depend on another, namely /vdu. 
For example, if we had not known the integral of a™da, we 


would make x == u, dx = dv ; and we should have 


== mil _ —fe. mada = aml —mfamde ; 


funds = 
et "er transposition, we have 
; (m+-1)fardx = am", 
and consequently 
1 


Numerous instances of the efficacy of this method will appear 


hereafter. ; 
Itis called cutegration by parts. A similar method may be 


deduced from the form for the differential of a fraction 


ve) ve] Vv 
u du dv 
U Vv Vv 


from whence we get 


fog fFZ 


This, as in the former case, makes the integration of one dit- 
ferential depend on that of another; but it is not so generally 


useful a formula.* 
184. We will now proceed to fractional functions, and to begin 


with a very simple case, we will suppose that 
Ax™da 
1) Vat a STE 5. 
4 (an-Foy ? 
making az-+-b = z, we find 


—_— 
— 


2 dh) d 
dx = —, 
a 


= adu, farde = afxdx, and conse- 


* It is proper to observe that, sinced.au = 
quently that all such constant quantities may be brought from under the sign fof inte 


gration. 
Ane 


Oy et 
Gao 


THE INTEGRAL CALCULUS. 189 


ad 


and consequently h 


is A(z—b)mdz 


amin 


dy 


3 BI 


expanding (z—6)™, raultiplying the result by dz, and then dividing 
by 2", we shall have a series of terms involving simply powers of 
the variable and constant quantities, which may be integrated by 
the rule in art. (176). Let us take, for example, the case of m = 3 _ 
andn = 2; then ‘ | 


e 


Me Szdz — 3bdz+3b?2—ldz — b°z—dz ; ; 


A(z— dz. 


dy = 
Sah 
applying to each of these terms, the general rule, we have 


aS abe arie oet lee 
sf [F -atancin fac 


We will now substitute for z its value, and we shall find that, 
when | 
i. oi Az°dx 
4 (aa-bb)” 


Hi = § 3 (az-+b)? 8b (ax-+b)-+3b%U(ax+-b)-+b9(ax-+b)—23-+C. 


We might deduce, without difficulty the general formula ; and if 
we had 
_ Alarda+-Brrdu+-Catdz. .. +. . 


. eatin 


we may write it thus 


Aa"da Bide Catdx 
Y= Catt | (ext t Geto t &° 
185. All differentials, which are rational functions, may be com- 
prehended under this general form, 


| , Ud 
which, for the sake of brevity, we shall represent by > 


N ow, inthe first place, we may remark that the greatest ex- 
ponent of the powers of x in the numerator may be supposed to be 
less than that of its power in the denominator ; for if it were not so, 


190 HE INTEGRAL CALCULUS. 


by dividing the numerator by the denominator, and calling ¢ the 
quotient of the saab and 7 the remainder, we should have 
at 


=fqdx-+- Ppa ; but g being a rational and integral func- 


-tion is xz, the integral of gdx may be found by the immediate ap- 
piication of ee rule in art. (176) ; and there will remain nothing to 


j find but Re: TT? a quantity in which the function r is of lower di- 


mensions with respect to a, than the function V. 
Let us take for example the rational fraction 


Px? +-Qr?-+- Re+S 


Ni Pe si ee a 


om Qe+Re+s’ 
dividing, at first, all the terms of the numerator and denominator 
by Q’, we shall have 
aE a x 


wet Q ae 


making 


we shall have 


- P's? +-Q'2?-+- R'r-S" 


ee ee eee 


o2- R" ae +S” 
We shall effect the division in the following manner : 


P'x®--Q’2?4+-R'ax +S" | ?-+-R"2x+S" Divisor 
—-P "73 — Ri" P'x? — P'’S'’s | P’ PS + M Te, 


istrem. (Q’—R’P’)a?-+-(R’ - P'S” )z-+8" ; a 
let us represent by VM and.V the coefficients of x? and > reaeiey? 
the 1st rem. becomes Mx?-+-Na+5", 


SMe’ — MR’'s~ MS", 
* ee 


and the second rem. (N—MR")x-++S" = MS”; eis 


we may represent this last remainder by Ka--Z, and then we haye 


s 


pits 
eae 
¢ 


THE INTEGRAL CALCULUS. 191 


Pr te = (Pratalidet SEEM, 
and by integrating, we obtain 
Faget ta BE as fe 
thus the question is reduced to the integration of | 
_(Ke-+L)dz 
22+ Rs +8" 


185. It follows from what precedes that, whatever may be the’ 
rational fraction which we consider, the most general form which it 
can assume, will be 


This may be resolved into as many fractions, whose numerators 
are of the form x“dx as there are terms in the numerator, and 
thus the problem is reduced to the integration of a differential, 
whose coefficient is of the form , . 


Ax* +-Bat! Foe tCe. 
the exponents being integral and positive, and a >a. 
186. Such a fraction may always be reduced to a series of frac- 
tions, usually called partial fractions, each of which must come 
under some one of the following forms : 


Mdx Mdx (Max (Me-+N)de (Ma--N)da 
S sppbeaae volar aa? aah: ‘(x*--a2)m 
If we make the denominator of the proposed fraction equal to 
nothing, we shall form the equation 


Pant Qart+e. ww. Rats = 


and supposing that we have determined all the roots of this equa- 
tion, we may represent them by —a, —b, —c, —d, &c. by this 
means, the first side of the proposed equation will assume the form 
of a product of x factors 


oe, z+a, c+b, x-+-c, «+d, &e., | 
these factors may be real or imaginary, equal or unequal. 


; 


;* “oe re . my 4 babs Pia a a 


192 THE INTEGRAL CALCULUS. 


We shall commence with the most simple case, by supposing 


them real and unequal, and, in that case, the proposed fraction 
may be considered as the sum of the fractions 

Adz Bdx Cdx 

t-te’ 2+b? ate’ sf 


whose denominators are the factors of the denominator of the 


proposed fraction, and whose numerators are undetermined con 
stants. 


187. We will suppose, as an illustration of the above, that the 
differential which it is proposed to integrate, is the following : 


pa ip 
a?’ 
and, since we know that 
a°— a = (x-+a) (4—a), 
we shall suppose 


adx is A B 
GanGra) Wiest oie 


Aand B are constants, which it is necessary to determine : for 


this purpose, let us reduce the second member to the same de- 
nominator, and we shall obtain ~ 


SS as 
—— — ——————— 


(x— a)(a--a) (x —a)(x-+a) ze 


suppressing the common divisor and the factor dz, there will remain 


a = Ar+Aa+Bc—Ba.....\.. ~~ (2)¥ 


and, by arranging with respect to x, we shall have 


(A+B)x-+(A—B—1) a = 0, 


a having an indeterminate value, as the proposed differential sup-— 
poses, this equation takes place, whatever may be the value of x; 


consequently, according to thé method of indeterminate coeff- 


cients, we will equate separately to zero the coefficients of the dif= 
ferent powers of x, and we shall have 


A+B = 0, A—-B—1 = 0; 


these equations give 


| OPHE INTEGRAL UALCULts. 193 


By substituting these valudés in equation (1), we shall ape 
therefore ; sy 3 


and by integrating, we shall find 


adx . ag 
J zag = HO- 9 NEPA LC 


and consequently 


YES wa Wapo=1(S*) + 
x es ' 


a a” +a t-a 
188. For a second example, let us take the fraction 
a?-- bx? : 


ee Ags 
ar x? ‘ 


the simple factors of the denominator are x, a—2, and a-+= ; there-" 
fore the expression, which we have to integrate, is” 


ar-}-ba* 
—— ——— — dx: 
x(a —x)(a+-2) 


make 
ae--ba7 “A B AG 
-+d2 i a 


——_— di = oe ag of 
a(x—a)(x--a) “aE ae, aa whe 


reducing these partial fractions to the same denominator, and 
adding them together, we shall have 


Pb Aa Art Bart Br?+-Cox— Cr? 
a(a—x)(a-ae) =. at(a—ax)(atx) 2 


Therefore, since the denominators are equal, we have 
a +-b2?7= Aa*—Ax?-+- Bas -+- Br?-+ Caz — Cr’, , 
by transposition, © | 
(B —A—C—b) a°+-(Ba-}-Ca) 2-+ (4a? a") = 0. 


And by putting the coefficients of the several powers of x = 0, 
there will arise ' 
» BwA—C—b = 0, Ba+Ca = 0, Ao?—a° = 0: 

2C 


- 


ae 


Negation 


Pia) 


i.e decomposition, ant & 
+ | solution of ne Ae ieee | 
“ab 189. In what precedes = 
} . denominator of the proposed : 
be not the case, the ecompe si 
gay be effected under th Hoye me 
4 diately in cae 
since these two fracti ns 
4. AEB Fag 
ee a “ot 
if the denominator of the | 
r (2-+-a)", or, which amoun S| to the camo ting is 


roots of the equation in art. ( 186) are equal, i 
Py gee for this actors a partial fraction Of ad the Sea im ait 


sm | : 
Primi igi hat a ‘pS ae 

ne eee ae ; 
- é + , ‘ 


— Ts. eee oe Fe a % 
. ae ; ; - : * 


~ 
ie 
. 


pe INTEGRAL CALCULUS. 198 
we sho Id determine the dtieatcnis of its numerator, by reducing 
this and the other partial fractions to the same denominator, and- 
then comparing the sum of the numerators with that of the propos- 


ed fraction (187). 


We might then integrate by the rule in art, (176) ; but it is easily 
seen, that we may substitute for the fraction wae 


Pram 4.Q"a—2h ia uc _RetS"), 


ee ne 


Parties 3 i xe 

the expression - . ‘ 7 PN i ve 

“Adee bb |i acl Adx i a A'dx Mai 2 a og AO da 
(e--a)™ ) ae (era Pe, 

for, by reducing all the terms of this expression to ihe same deno- 

minator, the numerator which we obtain will be of the same form 

with that of the first fraction. This” done, let «-+-a = z, and we 


have wi vat “4. . 
Adz pipette Pe Pet, 4 “ 
ses Raa: ~ G=m)(a-a)r— ? 


we shall find, in the same way, 


: Was Adx i Pe 
» 7 fo-aym—* (2—m)(x--a)"— q 


and so on for the ae all these integrals will be algebraiéal, 
except the last, pasties ae = which will involve a ‘logarithm. 
_ 190. Let us take, for example, the fraction 


Qaxdx 


” (z-Fa)?’ : 


we shail have 
br Qax A ® aif 
\ (gta tay aba’ 


reducing the second member to the same denominator, and taking 
ee ee common denpramator, there remains 


a Me my ; Sua = + Maha 


f 
whence we a deduce these equations of condition ~ 


i dy gia A+1a = 05 “ oe. 
which give 


- 


"ier 


196 THE INTEGRAL CALCULUS, 
ee a] ' BS UR oy aan 2a, A =: ee rt ipl a wy 
ben Witt a eye wet Le i Pt 
and consequently we BA igh 1 ae & sh | 
Cards in ode de fo 2adz™ |. Yay! Oe 
‘ "} (afta ~ (pa)? * (ea) 


in order to obtain the integral, it may be remarked that, dx being 
the differential of z-+a, we could sure nee: Peas 


7 
% 


ice as 


integrating the first fraction of the oer member by the rule art. 
(176), and the other by logarithms, we shall obtain © 


Siuedeahii2ah Uadllin, (> va 
Oe, ay pe 3 ee 


and, by mi for z its value, 


Tei 2a" a it a 
Je i Qalfxta)aCoi) ie 9 
91. Let us take fora second example the fraction 
ada 
- 3 Oo 2—@ gpa “ . 
the factors of its detominatar are easily discovered, for it t may be 
put under this: form | Meng AF ‘ hs 


ae 
i ~a?\(xd) = (a Fa) —a) ed) = Fes ON : 


and consequently, the proposed fraction becomes “a 


a" de 4 wit ees . 
Ea Aexa) * Lal eFa) 
Let us suppose therefore: { he : y if 


x? Ye 2 * 4 a bi, Bo ie: . y 
(2-@)'(2ta) ayaa oe We ith ; 
hye § ‘ A Oras % ach fs 4 
by reducing the second member to he same senomniastouie ob- 
tain x 3 al abe i 


2 a. _ te Pa) ae (2 ~ 0!) + Baa a)? he : 
(2-a)'(a+a) ‘ (c—a’ —a)(a-ba) i ae 


veloping and equating the “coefficients of the like powers of iv, 
we shall obtain these equations of condition : | 


oe 
“eye 
ry 


PHE INTEGRAL CALCULUS. 197 


pa 


| AEB = 1, A—2Ba = 0, a —A'a?4-Ba? = 0. . -- <2). 


if we ‘multiply the first by a”, and: add the result to the third, we 
shall have 


a} 2 Bat = a’: 
adding this to the second of equations (2) multiplied by a, we find 
a? = 24a, and vel = 14; 


by putting this value of .4 i in the second of equations (2), we ob- 
tain ' 


B=}, 
and consequently the first gives 
: ae 


by means of the values of these constants, equation (1), multipli- 
ed by dx, becomes 


dey ey ade Bde dx 
(2—a)(a--a) 2(a—a)* © 4(e@=a)' 4(a-Ha)” 


: adx 1 
In order to integrate - aoe ; make z—a, = z, and this expres- 


2( 
,* ed adz az—*dz 
sion will become -— = sere Floss integral is evidently 
az— eae a 
wiry Fey Matte) ” 


therefore 


“2 dx | : 
) aerspreeor i +3e—a) 41, Og 


192, A similar mode of operation, may also be employed, in'the 
case where the denominator of the given fraction has, several fac- 
tors of the form (2--a)™, (x-+-b)", &c. Let, for example, 


i ae : ’ 
Bi adx ada 


 @a1 =e Fy 


by paiBre den a bai a | 
aaerFT EF aso tate 


€ 


i 96 THE INTEGRAL CALCULUS. 


which expression, whee the fractions in the right hand member, are 
brought to a common denominator, will become of the form 


a PPR cal 
G>T @+1? 
Al) +1 A(x — 1)(a+-1)P+B(e—1 wen Dari ICT) i)? 
 @=1p@Fiy 
And, therefore, by rejecting the denominators, which are equal to 
each other, and reducing the numerator of the right hand member 
to its most simple form, we shall have _ ‘ 
a = (4'+-B)2°+(A+4+B— B)ai+ (24-98 —B)a- 
(A+B—A-+B). ‘ 
Where, by comparing the coefficients of the like powers of L, there 
will arise these equations of condition : 
A+B’ = 0, 
A+M+-B—B' = 
2A—2B— A’ -- B' = 0, 
A-~A+B+B = a. 
From which equations, when resolved in the Gaudi way, it will be 
found, that 


‘alene” 


A=1la,B= ge A’ tate B= 1a, 


By means of the values of these constants, the proposed differen. 
tial becomes 


is dx dz . Ae daz 
ie} GIP Gs) weal Bahl ‘. 


By integrating the first two terms of this expression by the me- 
thod in art. (189), and the other two by logarithms, we shall find 


Sx adx 0} aap ON Het) § +6 


193. If the denominator contain i imaginary roots, there must ne- 
cessarily be a factor of the form 


of Qan-b eet B? ; 


since the imaginary roots of equations always: enter tuto them i in 

pairs, the product of each pair of which is real. yw 

If we make a--a = z, the form becomes 22-++82, which is the 

“required form. If there “a m pairs of equal i imaginary roots, there 
will be a factor of the form 


w # 
; 


% 


hi 
_ THE INTEGRAL vaLcuLgs. 199 

Ts. ween ; 

Ph _ This, as before, may be reduced to the form (2? +6? Ym, 


' To the simple factor a°Qex+-e?-+~*, there will correspond 
a the partial fraction 


#, 


a MeN 


"tt j — e” 2 (a-Fe)?-+B? Ce 
: and to factors of the Loctad kind, the faition } 
Ye Gate eer bbe hy oaths +Y’ dx 
a (P+ 2ax a? 8?) iy 


but to facilitate the integration and to preserve the analogy with 
the formulas in art. an, we may substitute for this last the fol- 
lowing expression — ! AN i Pn 
3 — (Mx-+N)dx Meat \da " | (MO neNC "de 
: Cai ig Eee (4-07) 
The coefficients of the numerators may be determined in the 
manner we have indicated in articles, (187) and (188). © 
194. To ‘alearate the fraction 
. Ma +N 
pe de ; 
L7+Qaer-tart eo? 
or, which amounts to the same thing, the fraction 


Mz+P 
| Free 
W. whese sna, HOR = NMa: but 
Mz+P buy Mzdz 4 Pda 
a qa, ae eae ae | 
The first part of the second member of this equation can be 
io maearated uy ty making ane =u, we have zdz = 
“30 “, which gives A 
. | ve 
A Med2 Madu * 
tae oes . lia ys iyi 27-1 2) 


<5 Mif(e+-2en-+ a? -|- 87) 


_ With respect to the second part, if we make z = 6u, we shall 
* ‘have * | 


*. 


sad as a ° aoe Bic. ket 
+ e a ‘ - > 
Vs ‘ 
as 
° 4 
¢ 
4 
200 THE aNTEGRAL CALCULUS. 
" Pdz wD case PCr ' lle 
sis * ite aT es 2 5 rg ie 7 : ey . or . 
zB OB ge ee ae eS 
ir Ss ‘ee AB Nae ae So re ; ee 


~~ is the differential of the 


: < ior all A 
but we have secn in art. (44), that ie 
arc whose tan = 2 ; therefore 


P du 


a ote 
aC ae mm arc = eae : 

eh. 

“Bp 


are ( tan = a)+e -+- const. 


Adding together these two results, we shall get 


x 


cs Pe Nie de Ba a), 
— a aT = My eg are (tan ait : 


f' is prep to bithent! that the sine of the are mhose tangent is 
, Hi. ' he sie 
is and its cosine ——— ; for this pousiira- 
Fe TeERY SEER’ 
tion affords the means of presenting the proposed integral ‘under 
different forms, by designating the arc by its sine or its cosine. 
When we replace the value of z, we have 


. 


dy (Ma-+-N)dx 

e. hi mia | 
= My (a+2a0-te ae ( 

195. Let us take, for example the fraction eae ¢ 


ator ee abba 
Pe dN (c— Deine 


a°-ba-+-1 being the product of two imaginary factors, < as we can 
readily see by sdsolirting the equation ad = 0. ek ae: 


Ati Sih. Pa me AS tpt | 


reducing to the same denominator, and operating as we have indi- 


cated in the preceding article, we shall find | tye 
| jie air u= ~ a-eb Ne b—2a 
gee 3 ee 3 he 


let us decompose the factor ehetl into simple factors ; by « 
comparing it to the expression "x?--2ac--2 +2, we shall hate 


+ fF - 
rc 
: THE INTRGRAL CALCULUS 201 
, ie, Qe = 1, 2?+-6? = 1, 
. ey 
and consequent! 
% . 4 y of 
j , 3 
ut ves iS) — 4) 


substituting these values, and those of M and WN, in the last equa- 
tion of art. (194), which gives the second part of the integral, 
and observing that the first is 

Adx a+b 


aa, at fe 
a—1 i ) ay. 


we shall find —~ 


. bx)dx_ b ors eee 
SR OY VED 


fet. ; 
oe “ are } tan == LE i ior ‘4 
196. To integrate the differential 
Oe uate 
(a? 20x a°-+-@7)™ : 
we ‘shall at once mae a--« = z, and L—Ka = M; by this means 


we shall only have to find Vie: (Kz-+M) dz 


(e+e) ey which may also be written 
thus : 


zdz dz 
| Tere Verir 
The first part is integrable immediately ; for it is obvious that by 


: du 
making z?+8? = u, we have zdz = io and consequently 


fee zdz fis Key 
(Arey af au” ITE. —m)’ 
By. 

and the integration of _the second part we make dependent on that 
of the formula eye 
is pe by unity than j in the first. 

97. To integrate the second part, it will be necessary to have 
recourse to indeterminate coefficients. Let 


= x dz 
2 Lb. 


in which the index of the denominator 


oe! eee ee ees ee Og eee. 
I oe P nal a) rr  S E a , « : ; 
Ae : ’ * ~~ 7 


202 i's THE INTEGRAL CALCULUS. 


K and L being indeterminate coefficients, whose values may be des 


termined thus. Let this equation be differentiated, and the result © 


cleared of fractions, the factor dz being suppressed. Hence we 
find 


1 = K(2*-++6?)—2K(m — 1)2?-++ L (27+). 
Since these quantities must be equal, independently of 2, we have 
Fai (K+L)B, 3K4+L—2Km = 0. 
Hence, determining K and iz, pad substituting their values, we find 
Stes racer a 


Sa ee ED). 


‘This formula furnishes the means of depressing to unity the index 
of the denominator of the proposed fraction ; for if we put m—1 
in the place of m, we shall find 


omens 


i apa = 3a TE “ey? 3m —2)8* 


a. i . 


nian this value in the same equation (1), there will result 
ye OP sige: Zz path 2m —3 A With 
Mate Te (m—1)6%(2?-+p2)™—1 A(m—3)( (m—2)(5* 

z (2m— -3)(2m—5) 
ae 4(m— 1)(m—2 9 eu See oS ee (?). 


to 


dz ii. 
We may obtain, in the same manner, the value of Pi eam Bajn—2" 
by changing m into m—2 : # We were again to deduce from equa- 


tion (1) the value of fia a 
dz 


of f= » which would depend t upon Se 
ay. 


—, we shou ppiain a new value 


a —~,. And dite 
ihe process may be pune until the A. of a? (9? shall be 
reduced to unity, in which case the integral i is reduced to that of 
art. (194). . 

We here see the or igin of a method of integration, as fertile as 


it is elegant : it is that by which we pass from one integral to ano- 
ther. 


ae 


Le 


THE INTEGRAL CALCULUS. 203 


in comparing the results of the preceding articles we shall un- 
doubtedly have remarked, that differentials which present them- 
selves under the form of rational fractions, may be always integrat- 
ed, either algebraically, or by nieans of logarithms or circular arcs ; 
and that no other preparation is necessary, than to decompose them 
into fractions whose denominators are either binomial or trinomial 
quantities. 

198. We shall now V yueceud to give some examples of the ap- 
plication of the rules explained in this. section, for the integra- 
tion of differentials, whose coefficients are rational functions of the 
variable. 


ei adx 
3 ats ee ae. 25 = (1? 
Ex. 1 tee a BeLe ince « x+6 (x = 2} 
x —3); hf 
1 my. A fam B 


" aert6” yaa vas’ 
= (A+B)x—(3A+2B), 
- ALB = 0, 3A+2B = — 
a = 1, B = 


and consequently 


adx  _— g—adz, padxz 
Sf: mee fast hee = 2 HC) U(x —2)} ; 


ae Qe en ; 
panera (8 th ty 
(2 —4a)dx 


27—a—2 
2da _ ae 
fudz ={-— a/ ey 5 


bo sate =~ 2§(a—2)+1(a+ 1) } =—2I.(2°—2— 2), 


Hence 


Ex. 2. Letudy = 


yee 2. 
mt —«-+ 1)d : tee 
Ex. 3. Pa = aor: es The denominator in this 


case may be resolved into the factors «+1 and 2°--1, and thence 


nas 
ye 
\ 


¥ It is to be understood that the arbitrary constant is omitted in these examples. I 
must of course be supplied in particular cases where it can be determined, 


204. THE INTEGRAL CALCULUS. 


| , de ada da: 
hee Si feo J aqi! 
+. fide = 2 U(e-1)—H(2-+1)—} tanta, 


and consequently 


p(t ide _ @tIey |, 
ee ery tan~‘*2. 


| 29 +-3a7-4-3 
Ex. 4. Let ude == = ee therefore, by art.(196) 


— Qed a é d: 
a das Qaeda mm x if Qudzx * ax c 


HY Eee Gb ie eet 
And since 3 


pointe Qadx a 
(x*-- 1)? seri od Gp = ~ eal 


also 
Stay .. ery Ser 
Sea = ry CE STy fae 


— —1 
Le a1 tanta, 
Hence by combining these results, we find 


_™ gag Te - AO" 15 


> cde 
Mig ree Sy ear be) 
6. wide a? ag? , ore 
Ex. SE Sh Wheat BT Be 5 Mate) 
x"dax __ 2a? 2a... 
Ex. 7. Sie rt) =e 5 )e The 7 Katee): 


x'da saa Qa°x? 4a4 
cae Be OD at = G-5 ate) 


ey Sees eee ee a ae 


THE INTEGRAL CALCULUS. 


ak 
l RES 
tas Hab: 
ee, se ieee): 


f= 


> ade 
: Satigs = | — (5+ ora) ae 
xd. “3% __ bate 9a 
Be: ° Sarin Nb TB ab 7 Vice bx)" 


at = U(a--bz). 


xd te Zax , 12a°e . Gat i 
 } ue fori a a +5): 


a 


(a+ bx)? 


bs ‘ 
mun l(a+-ber). 

| - +6 

Bx 12 feta : walla): 

Ex. 13. fae=- = to A ei). 


id 1 1 ab 
a «pein. a(a--ba) a \(er)- 


—24 1 a-+-b 
oe 18 foray oak Sade arate rie) 
3 da 1 fears 
Ex. 16. Saray =-(— ft ve etic 


eh Ee) 
Ex. 17. hab. = st mn 


a onl 
an J (53) = a 6 sv C353) 


a He sec WG) : 


a 


ade 


Ex. 18. Let udx = ae ; if u-+ba* ==, we shall have, 


--- > = — = (see the bs Ex.) 


bs 
Csr 


NSO Teag coy Pee eee Oe, fg wh, ee aN 
“eg Sa) aaa iis 
hive ( 48a yY 


206 THE INTEGRAL CALCULUS. 
es MNS 
eon 
x? dar ue Va >a t 
AG bp berry 
weda: 2? a preeita 
XY,” Obey, 
ge! dx wy oc? ae a? nds 
xi dx set Maat a ory 
Ya oe Tp > tae 
xeda 2% ax? ata dix fs ee egies 
i a BN a hae 
cide CS. Canales | 8 . eda 
X 66 46? ' 953 b3/ X | 
xedx 27 axe a®x3 ase. at pdx 
aa ae td | 
dx 2 Zea--b 
{x. 19 ps Et IE arch Oa! ew sD BS UL, a TE ies . 
ry abr ex? / (4ac —6?) wy a/(4ac —6b?) 
dx 1 («ta 
Ex.. 20. a ee ee ee, 
i kere ar bom ae 
dx 1 I Cee ed 
ax, Se, ThE adh ty pall 
5 S eeaectoe (6—0) (a6) (6—a)? atb 
Pp COX eo . 
. 22. f ——_—_—_. --- = ——. Se ee i). 
m? i fags roger) b—a, (e-r9) 
| xd b a pire 
ix 23. fj ——- ee ae oe ie 
a JS ctageney (6~a)(a-+b) aes ‘ipa 
x dt | i 
ix, 2a. 
Ex perenen as =a Oe =a py, ) 
b2 ~ 2ab age ha 
om a2 * ~l(a-+b). a 
hg Oe 1 i ) 
Ba 8S area 2 Oma eta ee 


, 2 - baat 


bay Noth! 


ae 


mm 


% 


. 
?HE aad CALCULUS, — 207 
Bil. raat 7 Gea eat) 
' tema Gr) - 
Ex. 27. f7. aaa mye aae 
(ere i » ie 
< ~ Gana! ear b/" pi mn 

ae Sega lemas me ay 

ace a hair cme al 
Sie * estore er - ae iy Sit 

if aCe) U(a-+b) — on (= let). 
Bx. 30. fa 75 = 5 2) 
Bx. 3h fates = (SS). 
Ex. 32. Soa Ti = yt = omg 
ee 8 See s5it Aca ve Fan ° a 
Ex. 34. acy a+ 575m a a 3! he : 
Bx. 85. foes =F wn 5-51 (2). 
Ex, 36. a ea te 8 (a5 

+1 ea as are (tan = 2) + 6. 
‘ 


” 


POS THE INTEGRAL CALCULUS. 


> 


SECTION IL. 


‘The Integration of Irrational Functions. 


199. The ‘teste of diff e srentials, of which the coefficients 
are irrational functions of the variable, i 18; in ‘general, effected by a 
transformation, by which the fonction i is’ ‘rationalise if 
formations must be suggested by the expertness Pa eddies of the 
analyst rather than by any general rules. Our knowledge in this 

part of the integral calculus is considerably limited, and there are 

numerous classes of differentials, the integrals of which have ne- 
ver yet been assigned under a finite form. In the present section 
we shall treat of the principal irrational differentials which have 
been integrated in finite terms. 

We can always apply the preceding rules to the integration 
of irrational quantities, consisting of one term only. Let us take, 

for example, i 


@ 


(Fy ft— Ya") da 


b-+-3/a 


it is evident, that by making z=2", ‘all the extractions indicated by 
the radical signs may be effected, we shall thus have 


. a 62°dz( 1 4-23 atedh sa 
? ih he 
dividing by 1--2*, there results 


x, # 


6} sasiade Uap eds > ete ee c, 
{ at 


. t +22 
whose integral is ee es, A I 
a8. (pla oiheeen 53 a, a 4 
— 6 3 th —-— — Z—-aA = ‘ if 


“and iad a3 z by its value Ya, we shall have : 


% 


| “aves cabs — S84 24/2 —64/2 
| ’ +6 are (tan = yd 


ee 


THE INTEGRAL CALCULUS. 209 


200. We shall first consider those irrational functions which in- 
- clude the radical ,/(4-++Ba-+- Cz’) only, and which ¢ can only appear 
Sader one or other of the forms: Xdx/ (A-+-Ba--Cz?) “and 


VW FA — ee z Gay” , X being. a wational function of x. It may be 
observed that one of these forms i is included i in the other ; ; for we 
may \ write the first as follows : : ih 


| a KR Bu-PCx*) “ea VCP BE ECat) 
and the , numerator of the resulting fraction then becomes a rational 
function. © | am 
Before we proceed to explain the method of making the ex- 
pression ,/(4+Bx-+Cx*) rational, with respect toe we will ib. 
the quantity 4-4-Bz-4-Czx’, under this form, 


Bae MS St 
(t+ eer) 


and then, for the sake of brevity, making 


; Ae B 
C= a rao dipe B, 


Xdx Jf (A+Bx+ Cz?) X (ABCs!) ve KAR Bs Cds, 


we shall have r 


OM (At BatCe*) = 7 of (at+BR2-+2?*). 


This done, if we assume ,/(#-$2-+-2") = x-+2z, and square 
both sides of the equation, there ge result a-- 6x == \Qee4-z?, 


which will give | es 
. . ae go 2" et 
lee sa 227—p" 
from whence ; , . 
z oo Bere 
J (AP Ba-+Cx?) = (ate) = (3 = 
2(a— petzt) 
i ee aaa — 
i uc) (22—p)° 
By means of these values, we shall transform the differential 
, _ Xdx | 
JAP Be Ory "@ 


into another of the form Zdz, Z being a rational function of z, and 
also real, if Che a positive quantity ; for if C was negative, +7 


25 
$ 


Es) 


t} iii 
, Fs, ih 
ray 


Be OORT aN Sis AOR OE eee 


4 


210 ' . THE INTEGRAL CALCULUS. 


would become imaginary, and the transformed expression would 


~ become so likewise. 


In this case, we have to consider J (A+ Ba — Cz). and making 


, B 
Cane ae = £6, 


it becomes y/(e-+pr—2x"). The quantity 2°—@x—e may al- 


ways be decomposed into real factors of the first degree; if we 
represent them by «—a and x—q’, it is evident, that - 


et Ba — 22s am (977 '— Bax ~—a) = = («—4) Cine ig 


Then making 4/§ («—a) (a’—x)}= (oan and squaring both 
sides of the equation, it becomes divisible by x—a, and we have 
(a'—x) = (a—a)z*, from which we find 


a2*--a __ (a —a)z _ 2(a—a )zdz 
Pan ae 2+) 
values which will also render rational the proposed differential. 


201. We will take for our first example the differential _ 
dx 


Ji Ba Cay * 


the first of the preceding transformations gives 


—Bdz 
y(22—B) B) 
Substituting now for z its value —x2+-4/(e+6x-+2’), and for 
#, B, and y, the quantities which they severally represent, it will 
become she 


nmeek SA-5 eC - ny/Ce (AB Cet) t+ const. 


a result to which we may give the form 


, whose integral is -) l(2z—f)-- const. 


~s— }- 3 aed’, Sealant 


} 0 is 
-——-+ const. Y 
mw or } 


Combining the constant terms, and observing that the radical fC 
is susceptible of the sign +, we shall have 


if: da. ual I pes aC gate Ob Yabba Ce ) 


WV ( FB ECRD MG. 


+ const, 


~ re - 


THE INTEGRAL CALCULUS. 2ti 


202, Let us take, for a second example, 


dx . % 
VY (AFBr— CH 


making use of the last transformation in art. 200, we shall have 


Ps 
“an 


* \ ris 


og 2 
WEE) whose integral is or (tan = z)-+ consf. 
Substituting for z, its value ae =, deduced from the equa- 


Py 
tion ae = = (2—a)z% and putting ,/C for y, we shall have : 


dx - °2 __ of (a'—2) 
I Tatar ~e™ ( tan = J(E=0) )+e 
a and a being the roots of the equation 


fA 
7 me — Lm 


€ G 


=O 
and c the arbitrary constant. 
203. If we suppose 4 =: C = 1, and B = 0, the proposed dif- 


ferential becomes, in this particular CASe, Fay? and the pre- 


dx 
/(I- —x*) 


ceding formula gives 


d. aj  ¥(l 
cpr ae =—2, arc (ten = ven) te ; 


_for a and a’, being the roots of the equation 27—1 = 0, we must 


take a =—1, and a’ = 1, to avoid an imaginary expression. 

We now proceed to show, that this result is identical with an 
arc whose sine = 2, the differential of which we know to be 

dx , i 
wre ay For this purpose it is proper to observe that 
tan a 
tan 2a = ——-, * 
1 —tan*a.’ 


from which it follows, that the arc, which is double of that indicated 


Has x”) 


in the preceding formula, has for its tangent —~~——“, and that 


a a a a a tt ee er 


* By dividing the first were. (34) by the second, we have, since sin @ divided by 
cos @ is equal totan a, Aj 
~tan tan d-- tan 6 
a+t-b ——., 
ba 1h T—tan atan 6” 
and consequently, by making a = 4, we shall find the above formula. 


to that of the arc of a circle ; for, by making at first x - 


21% THE INTEGRAL CALCULUS. 
Sales: / it is the complement of the are) whose tangents 


—, and whose gine is @. 


ix . 

“peng this last are by s, we shall have’ : 
f vial Soe si tte se 
Jaa) 


and eee oe 5 with the ‘ia Constant, there will result 


wn. 


mend const. . OT Ni tgtt 
> patie i an 


‘ye 
We Will also observe, that we may directly reduce the differ | 


ential 


2 dx dx oth 
SA (ATBa—Ca*) af (era — 2°) © 


we shall get 3 
dz ais 
eri 13.27)" 
j dit 
then putting e-—-16%==¢", and z = ill aA cee 
en putting @--1, 6 gu, we wi obtain yaad 


. . 1 ° 
whose integral is — arc (sin = #)--const. 
Y 


204. The integration of the expression —— may also be 


Toe =) 


effected by means of logarithms, a method which leads _ to very re- 


markable imaginary expressions for the sine and cosine. 


By com afin this expression with Lge = fhd 
x P S P o/ (A+ Br + Cx?)’ we fin 
A= 1, B = 0,C=—1; and the general integral becomes (201) 
1 rapes i " 
——_ 4] 3 ie) my ne. 

if we represent by z the are of pains —— is the rential 
rae 4 aA gee ie +9 

we shall then ahs ¥ ae 

. 00 

ny ‘ 
ie = iy! ra)(—1 +./(1—2? const. 


But if we wish that is ¢ should be nothing at the oat time 
with 2, we must suppress the arbitrary constant ; for, by making 
« == 0, the second member reduces itself to this constant, inas- 
much as lig. & 


ee ee a nS 
i ’ . 


: | THE InTBERAL CALCULUS. 213 
‘This done, and observing that if 2 be the sine of the are z, then 
4/(1—2?) is its cosine, the equation above will become 
z4/(—1) = Ueosz+4/(—1) sin z} 5 
and if we suppose z negative, since 
sin (—z) =—sin z, cos (—z) = cos z, 
we shall have, in this ease also, fs , 
| —24/(—1) = I} cos 2a J(—1) sin 2}, | 
a result which may be joined with the preceding, in the double equa- 
tion | 
tha/(—1) = lf cos z44/(—1) sin Zz}. 


Passing now in each member of the equation, from the logarithms 
to the numbers to which they correspond, we shall have 


et2¥ (+1) = cos 2,4/(—1) sin z, 
an equation which furnishes the following two, 
ve 7v (—)) = cos z-+,4/(—1) sin z, 
eV (—!) = cos aay sin z. fs 
If we add these together, we shall find 


ey (1) ¥ (1) 


cos z = — 
2 


and subtracting the second from the first, there will result 


ee (—1) — eV (1) 

cata us) 
These expressions are nothing more in reality than pure algebrai- 
cal symbols, which represent, under an abridged form, the series 
(2) in art. 58, of which we may be satisfied, by substituting for the 
exponential quantities e7V —” and e~7“‘—, their developments, 
formed after the series (2) in art. 55, by making c = 1 ; but these 
symbols, though we cannot assign their value, under any real finite © 
form, are nevertheless of the greatest use in analysis, and exhibit 
all the properties of the trigonometrical lines which they represent, 

By substituting nz instead of z, in the equation 


etz9 (=) = cos 22b4/(—1) sin 2, 


A: 
it becomes " ye 9 


LE RCT, ES ero) | Re hs SR ON OME yee aye MRSS oe AN WP (A 


{ 


214 ; THE INTEGRAL CALCULUS. * : 
etnz(—1) = cos nz t/a 1) sin nz. 
But we have also 
etn / 1) = (et ¥(-)" = (cos 24/(—1) sin z)"; 
_and consequently | 
(cos z£4/(—1) sinz)" = cos nz VY cr NZ. 


This equation leads to reat of great importance, which we shall 
develope hereafter. ' 

205. We shall conclude the present section with a few practical ex- 
amples in the integration of differentials, whose coefficients are 


irrational. 
Ex. 1. Let udz = Va Sy ; then, if z? = a-+-ba, we shall 
have 
udx = ride 3. fudz, = o/( a+bz). 
Ex. 2. Let udzx = Toe, aif z= at-t be 
taf (ab) ’ * 
2 via 
2zdz == bdx, andx = : ; : 
Qdz 
oe udx = ee’ 
which has been integrated in Section I. art. (187). 
dx 
Ex. 3. Let udz = Jat?) : then, 


fude = 12§at/(atckat)} = 241 fx (G4 a}. 
Ex. 4. Let ude = 4/(2*-+-a’).dz : then, if oy (a°--a?) = yar ; 


we have udx = ydx —ade ; ; 
o. fudx = fydx—t2". 
Substituting for dx its value and integrating, we find 
Syda = y+ yarly ; 
2. fudr = hin Bie tito o/ (aia) ‘, 
dx 1v /(a--bt)—/fa : 
“ier l 7 ‘Tak teas 


Tags = pillevirv(otie} 


keds 


Ex. 5: 


. be * 
2 
© THE INTEGRAL CALCULUS. ° 215 
= yy ath sin a a 
¥ ., adv Ng 
4 ix. 7. Pie cer at y(i+e)} e 
Ex. 8. bye = Haty(a2=1)}. 
dex J+2)- )=1 y  liad 
Ex. 9.. dane ae, aa : * i 
da Jf (i —2?)— bv 
Ex, ,1059) ee 
x. 10 aes U a 
Ex. 11, Exc 1) = sec —!x = fan —1,/(421) 
= cos ab = 1 ¢os 4a? = sin ad (z= 1) 
x % = x 
d 
Ex. 12. far i facerpia 2) 5 
%' ig 
Ex. 1 18 Jeeta = Uf 2e+14+24/(1+2-+a")}. 


Ex. 14, Let the integral of —— be required : 


if a-+-bx==JX, we shall have 


Zee 


eae ¢ y- avi 


SFE AGP sexton) 


Sox (; XP ae — = aX*atX — —c° ) ah 


IF 


2X, 


=G-; aX%-L— 5 tXt 2 a?X-ta! )-¥ 


ee eee 


* These successive integrals may be readily obtained by integration by parts, ait, 
183 : the reader will find this method fully discussed in the next oo 


216 


Ex. 15. Let udz =({——- 


THE INDEGRAL CALCULUS. 


ie dc)» 


ERY (a Tale)“ : then, if ae = “y 


en i (see cabatble 5, ela 
dx mes b dx | 
ote ret fis a 
Dy d ax 2a 4 ' ey 
dx _ ey a . 
P65 ame ae, + yx ert oye fag 
dx. 1 PB et ee 5b° op dx 
f a an Mg oder )vXx be eal ook 
de: » <7 850° | 356° 
SEX yk ~ 4ax* ' 240° — 9608n2 64a*x sare) vx 
350+ op dx 
tinea ax : 
" andy * 
Ex. 16. Let udz = Via+ba*) Sa | a+bz —— 


SF dx 
Sz xidx 


sae 


Ex. 17. 


4 =\F (see ex? 6, page 215), 


JX 
ra 2X a vx in 
“Wb 8b “JX 


= (%- =v x 
= (5 -se) Vx tend oF 


x* 8a2 
(o5~ Fee Tee Rapes) ve me 
Let udz = 


4ax2 
1562 


ada, 


2S ae, —e a OY. 
wees); if 1—a* = 4X, 


St ‘so = arc (sin = x), (see art. 203,) 
P fX + oi-0 ans f a rw 


> 
+. + . 7 
o 


~& 


rite INTEGRAL CALCULUS. 217 
7, 4 ae wed We 
A pe aX Ras 
i | + , é 
[oe aa hey 4 Fe oy ed 
JK “Sele © 27 FX 
Me a died (5. ai+3) Vx . “ 
Ser F aj esr avi 8 pa . . 
5d © 
fatten (bs w+ a Pay) 
64) 1 y de ee 
Le aie (ge ep 5 ate rte 16 a heee 


i” 4 


SECTION It. 


The Integration of Binomial iferentials. 
207. The general formula of these differentials is 
ag” —'du(a--ba")” : 
if r be a whole number, this formula can be readily integrated by 


art. (181); but, when r is equal to the fraction ° we shall have 


¢ Pr 
Or eer das. Fy MS ee 
whose generality will not be affected by supposing mand n to be 
whole numbers. 


+ Lp 
If we had, for example, x?dx = (a--br?)q, we may assume 


a == z6, and the differential would become 6z7dz(a-+-b2’)7. We 
may also consider x to be, in all cases, positive, inasmuch as in the 


P 
case in which we have 2"—dr(a--be—")7 we may suppose « = —, 


and the result of this substitution will be 


i 
—2-M—ldz(a+b27M . 
2F 


218 HE INTEGRAL CALCULUS. 


To find in what case the equation (1) may become rational, let us 
make 


PS ‘ a-+bz” = 2? d a are ne SP Rey 
or, which amounts to the same thing, ie 
¥ i 4 
(a-+ba")q = z, ae 
and consequently 
: Ww 


a | 
(a--ba2")a = 2PH. tw o ~ ge 4 (8). 
Differentiating equation (2), we get 
al i Sans gzt—dz Shh Ee, FOL. to Adie (4): 


the same equation being resolved with respect to x, we shall have 


& 


‘ ; 
21 — a\ ~ 
at rs 
therefore, by raising both members of this equation to the power m, 
we obtain y 


re 
ae 


aes — th AR 
gh = — A 


differentiating this equation and pei by m, we find 


m 
hy er 
m—l a =< i (- )" q=!qz: 
‘ @ nbd\6 ‘ ; 
substituting, in the equation (1), this value, and also the value of 


(a-+-bx) y given by equation (3), we haye, finally, | 
LY is eR ae a 
n b ' 


° ° . : i i ee 
This expression is rational when — is a’ whole number ; for then 
‘ 07 


an Zl Oh , } 
the exponent, “aie 1, of 7 Is evidently a whole number. 


Let us take, for example, the expression _ 


‘ 


THE INTEGRAL CALCULUS. 219 


v(a--ba®)3 de 


in this case, we shall have ‘ 


eas m—1 = 50rm = 6)n = 25 


consequently t the condition of integrability i is satisfied. 


Substituting these values in the expression (5), we shall have to 
integrate 


wh 


3 ' Sze 3a 3a? 
ea d pth a ah Pear : 
ana —a)*2"dz =~ ope le — pa? et op ae tae ) 
therefore, ) 
e 


32"! Bazt 3a°2° 


2 tal 1 
fat (abut) 8de = ie apt ioe + C3 


we will substitute afterwards in this result the value of z, and we 
shall have the required integral expressed by a function of x only. 


208. The expression. x™—lda(a +bx")q is susceptible af another 
form, by means of which we will obtain another condition of inte- 
grability ; for this purpose, bd us write this expression in the 
following manner : 


te “40 zi : 
and, since : 
fewiecayit 


this expression becomes 
np P np 
— ya o— ao ——Il 
am—T4 (<4): dx =x" “40)4 dx 
x 


Ewer a Po 
—=aze2  (au—"-+b)1 dz. 


Now, by the process preceding, the last of these expressions 
may be made rational, whenever 


\e ee 
a , - 
% 
220 THE INTEGRAL CALCULUS. 
1 
n 
m-p—e ’ 
she m ike 
as or its equal yd 
n q 
es Mi 
is a whole number.” ' 
Let us take, for example, the expression | 
* 4 Th 
> \ cidagy(a + to . e e © e e ( 1) 9 
\ 
which may be written thus ; bis 
‘ 5 ah oe 
a°~* de (aba?) : 
we have , 
tt ” e 


m= 5 a= 3, Pp = i, q= 3, 


‘and consequently 


oop aes 
th ces ye gl eee 2 
n ba Sow 
therefore this quantity is Pitegrable. ma ‘, h 
In this case, we shall have (208), sl | 
ate ii 
z®— (a+ bx? )ede = (4 ay rig: 
£ ay ie 
~ = (Gu)! eR een Ne, 
making az~°4-b == 2°, we find» 
” " Vv i ae 
Ogi = git a 
or . = 6 * 7 
; +. Tt . 2A) 
Gay at tty 
the last of these equations gives 


THE INTRGRAL CALCULUS. 221 


multiplying the corresponding pengh ort of the last two equations, . 
we have 


» 


this value of x°dx, and that of (ax—8--b)3 being substituted inthe 
expression (2), we shall find ; " 

4. 1 @28dz 
an expression which is integrable 4 the method of rational 
fractions. 


209. Since it is not possible, in olay case, to integrate the ex- 


pression N 


P im - 6 
bern a™—dx (a-+-ban)?, | is 
the method of procedure which first suggests itself, is to endeavour 


to reduce it to the. most simple cases aha it can. include, as we 


have done in art. (197 th respect to 
dz 


Pa We shall effect this by the assistance of the re- 


-) which is rédiie- 


ed to 


mark in art. (183), i in which we observed, that fudv = =uv—fodu ; 


_ for if we decompose the quantity x™—'dr(a+-b2")! into two fac- 
tors, of which one being integrable, may be represented by dv, and 
the other by u, we shall make the integration of the proposed ex- 
pression, depend upon that of vdu, which, in certain cases, will be 
more simple than the given differential, as we shall proceed to 
show. 


For the sake of abridging the results, we shall write rin the 
place of a supposing r to represent any fractional number; the 
formula will then become 

21a (a-+b2")". 


Among the different ways of resolving this differential into factors, 
we shall choose that which diminishes the exponent of 2 without 
the parentheses thus 


ay ea" —* a" de (a bx)’. 
‘, 


ane THE INTEGRAL CALCULUS. 

By this means the factor «*~'dx(a-+ba™)” is integrable, whatever 

be the value of » (181): representing it, therefore, by dv, we have 
__(a+br)™ 
(r+ 1 )nb ; 


whence there results 


and w= xm%—" ; 


mk { n\r — snc 
falda (a-+-ba”) rs SET 


aa CEVA aioies, aaa et 
but 3 | 
fam—"—lda (aL bat)" = fam—"-lde (a4-bx")"(a--b2") 
= afam—"— da (a--ba)"-+bfa™—ldax (a4-ba")". 


Substilanns now this last value in the preceding equation, and 
collecting into one the fenns involving the integral fa™-"—'dx 
(a-+bx")", we ane , 


- oe 


bis ( pO) fam—ide (a--b2")" 


(r-F1)n 
hs am—n(at bar)! —a(m—n)frm—"—ldz (a--ba)r 
— (r-+i)nb 


from which we get (4) . ... . . . fam—da(a-+b2"), 
gn (aban)! — alm — ny ficm—"—1 der wale 


Oe ee i 


e 
_ It is readily seen, that since we may reduce, by this formula, the 
determination of fa™—'dx (a-bx")", to that of /a"—"—'dx 
(a--bx")", we may also reduce this last to that of /am—2-—'dx 
(a+6x")", by writing m—v in the place of m, in the equation (.4) ; 
then, by changing m—n into m—2n, in this last equation, we shall 
be able to determine fa™—*"—!dzx (a+-bz")", by means of. Giant 
(a-++-62”)’, and so on. 

In general, if r’ denote the number of reductions, we shall at 
_ last come to fa™~”""—'dx(a--bx")", and the last formula will be 


Sere nll (aeba” 


ama ban) "1 — a(m — r'n) rin) fam mde a-pban)” 
b§ra--m— (r= -1)nt — 


ad 


THE INTEGRAL CALCULUS. 223 


It is evident, from this formula, that if m be a multiple of n, then 
Jem—'dx (a-+bx”)” will be a finite algebraical quantity ; for in that 
case the coefficient m—r'n = 0, ne therefore the term containing 
fo™—"—1dx (a+-bx)’ will vanish. This result agrees with what 
we have already found in art. (208). 


210, There is another method of reduction, by which the expo- 
nent of the quantity within the parentheses may be diminished by 
unity ; for this purpose itis sufficient to observe, that _ 


Se'de(a-pba”)" = fx"—'da(a+ba")'—'(a-+ ba") 
= afym—t dx(a-ba")'—1-L-b fa! da(a--bx") aie 
and that the formula (4), by changing ‘m into m-+-n, and r into r=, 
gives y 
famn—ldx(a-p ban)" 
__ ea “bbx")"—amfon~hda(a-pban)" | 
b(rn--m) ea 


Substituting this value in the preceding equation, we have 
(B) a eee ee) fa Fde(a-bbn* 5 


uf (aba) ba® yt rnafe™—lde(a--bar)~' 
qi rtm 


By means of the formula ¢B) we may take away successively 
from 7, as many unities as it contains ; and by the application of 
this formula, and also of formula (4), we make the integral 
fam—'dx(a--bx")’ depend on fa™—-""—'dz(a+ba™)’—s, r'n being the 
greatest multiple of n contained in m—1, and s the greatest whole 
number in r. 


The integral Sat dr(a-+-ba)?, for example, may be reduced by 
the formula (.1), successively, to 


fatde(a-Lba®)?, frdx(a+-ba®)? ; 


and by the formula (B), frdx(a-+-b2*)? is reduced to frdx(a--ba*)?, 


and that again to frdu(a-+-ba’)?. 

211. It is evident, that if m and r were negative, the formule 
(A) and (B) would not answer the purpose for which they have 
been investigated : in that case they. would increase the exponent 
of x without the parentheses, as well as that of the parentheses 
itself. If, however, we reverse them, we shall find that they then 
apply to the case under consideration. 


a 
224 . THE INTEGRAL CALCULUS, 
¥F Pdi the formula (.7) we deduce : 
| foment a{a +-ba™)" 
¢ ihe 2m—"( ae bar)! — b(mpnfiemtde(a-h bay 
a(m—n) 
substitute m--n, in the place of m, and it becomes 
OCHRE CORON 08 ite otttnng AILEY 
xm(a-t bax) }— LAGisamins 2 peli Cue” 20 


atit 


a formula which diminishes the exponent of x without the paren- 
theses, since m--+n—1 becomes —m+n—1, when we put —m In 
the place of m. 


To reverse the formula (B), we take 


Jatcr'da(aqsbe" yo : 
egy x™(a--ba™)\"— (sn nr) fam—'da(a--ba")" ; 


THEA v 


then writing r-+1 in the place of r, we find 


(DYE chee a ot glen toi ee 2 iM ih ome aE atbany 


a (aq-ban) (mpage r nen) fom—!de(a-tbany 
~ (r-+ijn 1)na 


This formula answers the object in view, since r+1 becomes 
—r+1, when r is negative. 

The formuls (4), (B), (C), (D), cannot be applied when their 
denominators vanish. ‘This is the case with the formula (4), for 
example, when m = —2r ; but in all such cases the proposed func- 
tion is integrable, either algebraically or by logarithms. 

m—1 

212. Let the formula be f= yar a he 2 seyvhere m is a whole posi- 
tive number ; we find by the fortiula making a = 1, b=~—1, 
n=2,r=—i, 

Sz eed ae aol, Soke —x*\)  m—2 pax sdr . 


Vv (=z?) m—1} m— | o/ (Lim?) ,. 


writing m in the place of m— 1, it becomes 


eda atl /(1—2?) -m—1 2 om 2dr 
pn Bie sc uel I sag ~ 1 pe 
J /(1—2’) m 


m (l= x?) 


w 


HE INTEGRAL CALCULUS. 996 | 


if we give tom successively different values, beginning with the 
odd newt we shall have 


e adx . yO , | 
a (I -at)t const. 5 ‘ef oe. 
ae, (1—2") . 
; a'dx 1 , 2 uals in 
Bae tae Ae S Fiz a 
feat 1 
es an sO we fogs = 
ce ‘ 
From which we find | | i a 
Ss =-—-4/(1— 2?)+ const. 
ede 9 
IRF (0+ f/ (1— 2”) -- const. 
a 1.2.4 
Iz = ae a = (— 2+ ; taba ne eked 9 \+e, 
Xe. 


The law of these values is evident. 
Passing now to the even values of m, and supposing m = 2, 
m = 4,m = 6, &c. we find 


cig. ee | : Lig, “see... 
Poa ay a Ota ae oa 
fp. «dt S x*dx 
eaten g Po tht Ig otis 
» xda : , ; vie 
tore 73 OBE. wigs 
&c. Xe. 


In all these cases, the aaa integrals will depend upon 


ops Be ‘ : 
dzaee; == arc (sin == 2)-+ const. (see.art. 205.) 


and if we represent this are by 2, we shall have 


¢ 


. eda 1 1 
a? yy ee a 
—5t vv (1—zx’) Ps A. const 


J Ai — 32) i 


“ 


226 THE INTEGRAL CALCULUS. 


xu'dx ” ey 195 > il We Pda ee 
I Jase = —(j# on Ecole a Al -+ const. 


 adx he My Le oh e Fre 5 
Signer -(# +i76to a . 6 2) V(lma ine 76 

oe A+c, ' 
j ' 213, We now proceed to find the formule for those cases in 


which m is a negative number. We have then, by formula (C) 
art. (221), 


We Ws Re 
; sing 
tse xm, / (1 —2") +m—t e—mtldy 


a/ (1 —2x*) ay an a/ (12)? 
and writing —m, in the place of —m—1, it becomes 
7 dx a be Ay a ae dx 

u/ (1-27) (m—1) a1 m—1/ a2, (1 — 2”) 


We cannot here suppose m = 1, since that value would render 
the denominator = 0; we must therefore have recourse to the 
f dz: pie ss F 
Aner of N bsyzcm, which has been already found, (ex. 10, 


page 215,) from what has been said in art. 200. We may also 
arrive at this integral in the following manner: make |—a? = 2°; 
from which we have 


2. WT | Ae ga ene 
OA ey aes 
and consequently " 
x —dz 


? 


A 
an equation, the neared whose second member is 
* —9l(142)+9 10-2) en Ges 
and substituns forz its value, we hin: 
oh "e. "pe 1+ Ae (= 2%) t 
t we la, AT = sa ~a*). 


multiplying by lf (1—2"), the two members of the fraction 
ae under the sign 1, we shall get 


1S ee eae | 


ee eee 


= CELE), 


- THE INTEGRAL CALCUL us! 227 


we shall i then nally 
ws 
We (mente wid pm: + )F const. 


And making m = 3, m = 5, &c. we shall find ag 
“ hy dx tah ee aie | 
: ISIRER ce ty ae 1a") 

dx ida / (1 —2") cat iN hs i 


aaa a- : 7. 


dec ® hee. /1 _ 28) 5 dx 
Saga nay Fe. 6a ws. a a/ (1 2") 
Xe. &e 


Again, making m = 2,m= 4,m = 6, &c. we shall find 


dz . a/ (1 —2") 
Ie7a=y oan . -+-+const. 
igus 81-2’) 
fe on 2) RR 3g +3 fas wr, a?) 
f: dix +4 Vv (12?) 4 de 
Mea a) WTR | 5 Sy oe) 
&ce. &c. . 


From these two series of equations we shall be able to deduce, 
as In ‘the preceding article, one class of formulas integrated by lo- 
garithms, and another class which will be entirely algebraical. 

214. We will now give some examples of the rules for the in- 


tegration of binomial differentials. ie . 
Ex. 1. f- he =e F 3, (a-r bx)’ —2e(atbs)—e 
(a+bx)? 
, 2 
i oy CaS 
dx i : 2 
Ey.2 i Ba dt 
l +a) of VAN ] a4 ai 
” Sar 
se] AT ee ™ 
a 2 Wi l+-z)4-1 


* It is understood that the arbitrary constant is omitted in these examples. It must 


of course be supplied in particular cases where it can be determined. 


Ex, 3. 
wy 


q . € a 
, & 
Ae INTEGRAL CALCULUS. * 
d ae 
ine “4 =f vieth) fhe es n 
(a--bx)2 ° beabbeyE 
de 2 
x mg =f yetm—fe ‘ x--s all 
(a-+- ba)? b?(a-+ba)? 
xdz ; 2 ‘ 
ie . —/ (a+bz) sLle z° 
(a-+-ba)? aba)? 
dz 8 PA he 


“(aboet 3a 


(a a-tbx)# 3 


I dz 
——— ———o . ako é 214 ° 
tad ay (apbay” (see ex. 5, pagé ) 


Exs 7 


Ex: 8: 


Ex. 9. fax(atba)? 
: & 


Ex. 10: 


dx oy 
ee ee 
x(a--bx) 2 ich 


* I gf dx 
rae aS el opts)” | 


az 


—= H 
—- <== 


by 


14ba . 2672 
302 | a 


x(a--bx)? ax(a-+-bx)? 


e. dx 
pa errr 
S iy 


a 


care 1 


(abba) ? 


, (see ex. 5, page 214). 


b d. 
were. 


a(a-+-bx) + 


= s(t-tba) +a) 2,/(a+bz) 


(Ex. 5, page 214). 


aus ar Terry 


bay? ba)? 


3b2 pf, ? 
= aes (Ex. 9.) 


¥ * 
Ex 11; Let the intégral be 


shall have 


: if a+-br = X, we 


* a i g 
sf 
« * 
‘ "By vie INTEGRAL CALEOLUS. | * DDO 
Gee me (¥4 a ’ " ae 
f i 
x2 di 
pe = e inane) gt 
X2 + 


frie. bal (;x°- aX? 48atK-ba!) 


x2 
aidzx ; g 
if 7 = =( 5 ae 34.902 X2 4X —a — ee 


Ex. 12. Let the integral be Sx™da(a--bx)? > if atbr = X; 
we shall have 
$ 2X kX 4 


; 9 ¥3 ; 
! fedaX? = I X= ee 2X Jf X / 8 
: 9 6? 


t Vt / Xk 
a ae 


ee me .. 


exX3 7X 
hao 


frtdnX® Z (= XS aXi+— ex wa *) 
13 11 


5 6 9 ‘ 
fartdxX? ori (:¥-F0x4+ ar —5 aX +20 “2 x ef 


SECTION IV. 4 
Integration by Series. 


215. A series representing the integral of any differential may 
always be found by developing the differential coefficient in a se- 
ries of powers of the variable, and integrating each term of the se- 
ties after multiplying by dz. Thus, if x represent any function of 
vt, and 

X = Ax*+Bxd+Cr°+, &c. 
Axtt! Bybt! Caett 


rt ee aa 


SXdz = 5 


Pa 


230 THE INTEGRAL CALCULUS. 
y 


Although such a series is always:an analytical representation of the 
integral,,yet it is of no use in obtaining a value, or approximate 
value of it, except when it converges. It is proper to observe that, 


if, in the expansion of X, we meet with any term of the form *, the 


integral corresponding to that term will be Alz (179). 
216. To develope an arc $ in a series of powers of its sine. 
x oa! 
Let sin @ =<; ..d@= ———~. Let (1—2*)2 be deve- 
loped by the binomial theorem, |» 
i. 3) M3. 7 


an coe hg 13 tet as 6 eG _ 78. 
(lena!) "3 = 1+ ot sew a ace” ater e ee 


Multiplying by. dx and integrating both sides, 
6 Lig? PS ee A eres 
(Sites ts ae tee omer t gee 
which is the development required. No constant is added, since _ 


a = 0, renders @ = 0. 
217. To develope an arc ina series of powers of its tangent. 


dx ! 
= ly ° Loe —- & 7 1 : - 
Let @ = tan—'w; ..do ita Developing rhe by com 


mon division, we find : 
1 

ita? 

Multiplying by dx and integrating, we find 


= l—27?+2'—2°+2* —, Ke. 


e838 505 ee iieae 
— — ws —— —* os _— —_ & e 
oe Tagg eee 


No constant is added, since when @ = 0, 2 = 0. 


218. To develope a: an arc in a series of powers of its cosine or co- 


tangent. 
dx EN 
sf 9 = cos—r; 0. bs = ~——--.. And if ¢ = cot—xz, 
Abbe) ng 


d | ; 
ag = aria Hence the developments in these cases dif: 
x ; 
fer only in sign from the former two ; but since in these cases @ 
and x do not vanish together, it is necessary that a constant should 


be introduced. © Let this be C, .*, in the first case 


$= C— sin—'z; 


eth 
THE INTEGRAL CALCULUS. 


q : ' & ° us 
a .. Cos—'2z+sin—z = Cy, C= 3 
In the second case, also. . 


cot—!2 = Cotan 2; 2°. C= 


ty} a 


The sought developments are therefore. 


age x a 1.32 

onli Ses am 203 OPO ae a ae 
ee ee Maee ae ee 2 

OTT a vie Mie te) ate Sts ee 


219. To develope the versed sine of an arc in a series of powers 


of the arc itself. | 
dx 
es in—!a: .° SS 
Let ¢ sok. sin—'z; .. dg eins) 
= = a developing (1-42)—2 by the binominal 
(21)../(1— Ga) 


theorem, we find 


: ONO lg Pia a a 
ya eo eae we A 6 8 
/U=52) : 


Multiplying beth sides pylut 


= avs teeta, ee EE» taney 


No constant is added, because when z = 0,¢ = 0. 


z and integrating, we find 


wh 
Seater it & 


220. ~The most simple function of z that can be aes ina 


series, is rig which becomes, by . division 
x. 


hides sot I Mk — 
aa? a8 at 


+, Be 3 


from which we have 


dx ee ge 
SCP seal! Kae ee &c. + const. 
of 5 a 2a? +3 3 ts J 


but we also know that fz. = I(a+a); therefore 
a Qa 4b 


% 


¥eS 
#, 
93? THE INTRGRAL CALCULUS. 
x 8 2? x3 a4 
ata) = —~—4—~ +)... sal Be, const. 
( } a a2 Ez 4a? + aot © 


: Hong “ 
To find the nature and value of the constant quantity, we have only 
to make « = 0; for then the equation becomes /a = const. and 


consequently 


2 8 4 | 
i(a+a2) -la =1(1 1+ ~\= ae pat Xe. 


p 
221. The formula 2"~'dx (a+2")¢ is easily integrated by the 


P. 
expansion of the quantity (a+-ba”)¢ into a series, from which we 
obtain 

PY P_Q am bamtn (p ‘ie q)b? gt ron 
xr™—ld ic (a-+ban)\q = ag oe i ki dem iad teolera ag 
I (a“roee) m  gam+n 1. 29q°a? m+-2n 


gt *3n 


a a ~ 9) (p — 29)5° 


prs I 
Be al 3q°a? “m+4e3an ”? i f or const. 


If we wished to have a descending series with respect to 1, we 
must give the proposed differential the form 


Oy ae Po 
fg ee (b+ax-")q; 


“ Pp . 
and-we should find by expanding (6+az—")q, multiplying by 


mp? _y ; : 
z qa dx, and integrating the result, that 


(p—9q)” 
ir} qemty pa gem g 
pet qo mq (p— |)" 


fcom—dz (a-+-ban)g = 


en 


ea -q)a? gum j 
“tH ae. are pits yt c. ¢ + const. 

Whenever a and 6 are both positive, or gan edd number, we 
make use at pleasure, of either this or the preceding series ; but 
when q is even, the first formula will. become imaginary from the 


ene Sat. ; .) Ses 
factor aq, if ap be negative, which will also vil to the second, 


if bp be negative. 
222. The object of integration by series being to obtain approx- 


imate values of the integrals which we cannot obtain accurately, it 
is of consequence to have several series, so that we may be able 


f 


‘By 
a 


% . 7 5 is ae . . 
“DHE INTEGRAL CALCULUS. 233 


ie choose that Which becomes convergent upon the ashi of 


a proposed value of «. Those series which proceed by positive 

and increasing powers of x, or ascending series in general, do 

not converge, unless when 2 is very small; whilst those which 

proceed by the negative powers of-x, or descending series, become 

convergent only iehien ‘a is very large. 
r : is 


Let us take, for example, the differential by reducing 


amdx. 
ar-t-an : 


~~ Into a series, we find 
a i 
1 m 1 a” aca gon 
—_= PS ae &c. 


a"-- a a” a2” qi" 


andit will become, after multiplying by x™dx and integrating, 


x™da gmt i a_mpnpl gmt inti ( 
{= HBAS 6g og 


a “(m+ 1) a* (m-Pn-F 1 )a Cees -}-} a3” 
se 


Spe 


const. 


an ascending series, which converges very rapidly when # is very 
small. 

To obtain a descending series in this example, we must change 
the order of the terms of the binomial a*-+-a”, or we must put a, in 


} 1 
tne place of a in the development of aCe and we shall have 
| Tu 


1 1 a”™ ae” wih 
— = eed on rw +, &c. 
v 4x7n 


z+ a a” me 


and it will become, after multiplying by «™dx and integrating, 


£ boa: a ] a” 
2 x"-+-a" om ~ (nme Lik ere @ ew m— m1 )aen—m— “ 


Fo Th &c. + const. 
This series would fail, if any one of its denominators, which are 
comprised in the formula inm—1, should become equal to no- 
thing, which would be the case if aad were a multiple of n: in 
this case the expanded differential would contain a term of the form 


: 
a 


dx 
ama , whose integral i ipo Ml x. 


223. The object of the reduction of differentials into series is 
2H 


¥ . ¢ 


“4 
B 


234 THE INTEGRAL eAteuLUs.* : 


to transform them into a serie$ of terms, each of which is separate- 
ly integrable, and it is not omen neal for this purpose, that 


| eur 
all the terms should be quantities of the form Arn dre. 
If we have, for example, | 


ty 


me « dx,/(1 —e?x*) "¥ 
JOP ye , 


where e is a very Small quantity, we may expand ,/(i —e*x’) into 
a series, ponverming with great rapidity, inasmuch as in the propos- 
ed differential, x? is always less than 1, on account of the radical 
Vi ; we find, by the binomial pe hows 


él 
oy 


the series to be integrated is therefore 

| dx I ee Sl Rp be a 
SFP eg ee OS: 

Each term of this development comes under the form 


“i 
fii 


which has been integrated in pee Il. Ex. 17. | Substituting there- 


fore for 
“dx >» 3? a xtdx 
ae. ica (ae 
their values thus Hotere we find, by putting are (sin = x) 
equal to 4, 


if Sees cae 
POL en?) 


+38 vijay 
Vals ta} (ie on BD eoos a 


85 eget rag Mle aa et 


HAGiras Fad Saal ieee ge Mw Sal FO. heer -eeonst. 


iP 


. . > = 7 
THE yrrecaay GuLcuus, 235 


We might also treat ina similar manfer the differential. 


Ce cal PCE ial 
Vid-")@+ m) $f —a) of (ate) | 


Vv m2 ‘ 
by reducing into a series the quantity ‘ 


Va ay se yor 
{it is proper to remark that the formula 
aw du . i 
j v3 (2mu — u?)(n—u) } 


which is met with in some applications of this Caletlus to mechani-. 
cal pbleaiede itself to* 


Fond ia ierayy 


1 ’ 
by making 
nem 
m—wu = mz, and ae 
vit) 
. 4 ae nn 
since 2mu~u? = m?—m*a", n—-u = m aoe zi, and | 
| ak ‘ am 
re ws 7” 
» du=—mdr; ..° ws 
fs Pes dee ‘ ? , — mdz a 


A baat ae) a Se aia or +2)} odie 


ai 7 dz 

Jind ff 1a) (abe) 

224. As the development of integrals into series does not 
lead to an approximation, unless when the series are convergent, 
which does not always happen, it may be proper to take notice of 
the following general method of approxiniating to the values of in- 
tegrals by series. 

Let z = fudx-+-C, C being the arbitrary constant ; and let 2’ be 
what z becomes when x becomes a+. Now, since 


dz d?z du 


— = Us Ph ck EY Sons poten 
dr ee dz’ 


o> ~ 


¥ 


Sh 
4 ¥ 
ow 


236 | THB ANTEGRAL GALCULUS. 


oy eee aa ‘ 
and, in 1 general, " = —_—-. Hence, by Taylor’s series, 
dz" dan! 
‘ a yA d? sal 
th drs a 
f 
2 = 2+u. —+-—.—_—_ ++ —+.—_— &c. 3 
a pr aM 8 1 el ie Bt 
rd ‘ 
; 2 dhe Gu ie 
se “a Zs == Uu.—-++-—.——- -|-— —-+-, &e. 
1 dx 1.2) dx? 1.2. % 


The arbitrary constant disappears in this series, because it 15 
united to both z’ and z by the same sign. a i 

This series expresses only the difference between the values of 
the sought integral corresponding to the values x-+i and x of the. 
variable. Therefore the integral itself is so far indeterminate. 
But it may be observed, that, by whatever process the integral may 
be found, it isin this respect equally indeterminate; for the ar- 
bitrary constant being necessary to complete its value, all that is in 
any case obtained is the difference between the whole integral 
and its value when it becomes equal to the arbitrary constant. In 
the present instance, the integral is said to be obtained between the 
limits « and x--7, for when 2s 0, the series vanishes, and it gra- 
dually increases with 2 Ge beeen x+i,2 posuming some pro- 
posed value. j 

The value of the variable x, which makes the integral vanish, is 
said to be the origin of the integral. When the limits of an in- 
tegral are not assigned or known, it is called an indefinite i un- 
tegral. ‘Thus all integrals in which the value of the constant is 
not known, are indefinite integrals. But when the limits. are as- 
signed, they are then called complete or definite integrals. 

225. Inthe preceding case, if the limits of the integral be sup- 
posed to be « = a and x = 4, the value expressed ina series 
will be ‘ 


(bra) te pe he 2), mi sai 
At oe 5 CER ES 


du d2u4 
¢ dx? “7.22 


sf&ce. 


where .4, .1’, A”, &e. are what Uy &c. become when +=a. 


The series 


% ‘n ye, 2 
arin fasts ee Me 

is only convergent when 7 is very small, and therefore would only 
determine the approximate value of the integral between very 
narrow limits. This inconvenience, however, is remedied by the 


ae 


a 


&* 


THE PTEGRAL CALOULUS. 23% 


successive application of it. Letz, 4, 1’, 4’, &c: be the values of z, 
du d?u 

da’ dx” 

corresponding to 2 3 25, -2,, 4, A’,, &e. those correspond- 

ing to x+-2z, and, in general, z,, ‘A Ans A’, &c. those corres- 

pondie? to a-fut. . 
Hence we obtain the following series : 


&c. corragagnding tow; z,51,, 4,,4',, Re those 


_ 


awd 
yau2 = A. “tt wip ag &c. 


Ae 04d be. 2 
2,—-2,= Ay sta, rca fae rear &e. 
. Berit? A a +A’ ( @ 4A" & 
age a he) se fa?) 4 
: | A ia x 4p Say 4 & 
Ch han ed ola Lome Seon 
Let S(.A) signify ALA, + 4,. .. «- Aa; and let a similar 


symbol. express the sums of the “other Si bilicioni By addition 
we find . 


+ * 
\ 


jae SM pts); SU st, &e. 


This series converges the more vagy the smaller 7 is assumed. 
By these means we are enabled to integrate by series between any 
proposed limits. As before, let the limits be x =a and x = b. 


Divide buna by n, 
will be 


nas 


“=i. The value of the integral 


cited S(A):b—a) , SA") (b-a) 
n 1 2 Wears" 


which may be made to converge with sufficient _Tapidity, by assum- 
ing n sufficiently great. 

It is obvious that this method becomes inapplicable if any ex- 

ception of Taylor’s series be between the limits « = a and x =}; 


which is cicated by some of the coefficients becoming infinite. 
, 


és 


THE INT - plgiagion CALCULUS: 


oa nt iy 
pre S 


SECTION V.. 


The Integration of Differentials whose coefficients are Expo- 
nential or Logarithmic functions of the varvable. 


226. The integration of transcendental functions is effected by the 
aid of the several formule already established foralgebraic functions, 
united with some primitive formule peculiar to “themselves and 
derived immediately by inverting the rules for differentiation. These 
functions may assume such an infinite variety of forms, that no 
general methods of integration can be given; and, indeed, even 
were a Classification possible, these are many formule whose in- 
tegrals have not hitherto been assigned under a finite form. An 
approximation, however, may always be had by the method of series 
explained in the last Section. 

Ifu == at; .. du = actladxrs Hence 


ae nm 
fatdre— 
’ la 


this is the elementary integral of exponential furictions. 
927. A differential’ of the form Xa"dx, where X is an algebraic 
function of a", may be integrated by making a? =u; therefore 


ee: 
the differential becomes pecee but ude = =; . fAatds = 
; i 2 . 
af) F(u) du, which may be integrated by the rules already’ given. 
a 3 me 
By differentiating Xe,, we find hy " 4 
d( Xe”) == Netdx--erdX ; . 
and if a =: X’, we have a 
dx 
, d(Xe,) Ser X42’) dz. es 


So that every differential of the form e*F (z) dr, and in which F(x) 
consists of two parts, of which one (X") ) is the differential ‘coeffi- 
vient of the other (X), is easily integrated ; for in that case, 
fe?F(x)dx = e*X. An example will make this evident. 

Let the differential be « — 


e?(3a2-+-g?— 1) dx : a 


& 


THE INTEGRAL CALCULUS. * 239 
mow, since Jit 
® ~ ' Yagu 
guia (1° 7 # 
. gee) — 32, 
-. game? -olag | 


7. fer(3a?+- a3 — Idx = (x? — 1). 


_ 228. In most, cases, however, it will be necessary to integrate 
dy parts, and to establish formule of reduction by which the ex- 


ponents of the functions which are in yobs | in the differentials may 
be continually reduced. 


Let us consider the differential 
Ar xd: 


integrating Pypens we find 


aa? th 
fata dz = re ~ fa adr. 
a 


By successively substituting, n —1, »—2, &c. for n, the exponent 
of x will be reduced to 0, and the final integral will be 


fardx = — This process gives 


Neb F).... 1) 
(ayy 
the sign -- corresponding to the case in whet nis odd, and the 
sign — to that in which nis even. 


” ita) (n—2) ig 
“Tay? aa “eer: 


229. Ifthe exponent n be negative, this. series will not attain 
the desired end. In this case, however, by integration by iin we 
find © , 


4 


ne |) oo Oe 


‘ a® la dedar 


which produces a continual diminution in the exponent of z. The 
2atdx - 


final integral in this case will be : Sa This integral has never 


been assigned under a a finite form. it may, however, be developed 
ina series thus. By Maclaurin’s theorem, 


‘an P 


®40 HE INEKEGRAL CALCULUS. 


e ie /2 ate + Gy xi, &c. 


cy rags | 
Pe ek ot | 


+ J 


Multiplying by dz, and integrating, we find © 
e l l 
frSee +s Oe g* as ot, & 


me. e se 


If n be a fraction, it will be alo necessary to complete the integra- 


tion by a series. 

230. All the preceding observations. opp in general, ‘to the 
formula Ya"dx, ¥ being any algebraic function of a : for, let Xa*dx 
be decomposed into two factors a*dx and X; the integral of the 


first is a ; and we ‘consequently have 
Ob J 
| 1 1 , = 
SXardz = TA at hath Making 


dX = X’ ap dX’ = X"dx, dX” = X"dx, &e. 


and continuing the preceding reduction we obtain this series : 


1g nO 
[ee ia ae oes ‘or 


SEE ras 


| + aap 
the sign -- corresponding to the case in Sich nis odd, and the 
sign ~ to that in which nis even. The application, of this formu- 
la will lead to an exact integral, whenever _X is a rational and inte- 
ger function ; for in that case the number of the quantities X’= 
dX dX’ : a a” 

ae » A’ = pis , A” = —-, &c. will be limited ; the last will be 
constant, and consequent LX Maeda will be changed into Xo-» 


fartdx = Xe —-f-const. 


231. We shall now proceed to the integration of logarithmic 
functions : we shall remark in the first place that, if wu = Ix, du = 
d dz 
=} te a =lr. This is the elementary integral of logarith- 
mic functions. 

Let us take the formula 1 fXdee(Iee)", where Kis is an algebraical 
function of x. By integration by nite we find =» ° | 

- “ihe de 


THE INTEGRAL CALCULUS. 241 
| |  e(le) dz 
fXdz(lxz)" = (leyyfXde—n frye. 


Since X is supposed to be an algebraic function of x, the integral 
fXdz may be considered as known. 
Thus, when 2 is a positive integer, the above formula will, by 
continual substitution for n, reduce the exponent of the logarithm. 
232. But if n be negative, it will be necessary to integrate by 
parts in another way. Since 


x (lant! 
Stuy = OE, 
if the quantity X(/x)"dx be supposed to consist of the factors Xx 


and Co" ae, we find 


1 
Saxe = (layin + fly d( Xe), 

By the application of this formula the integration will be reduced 
to that of the formula /xd(Xx), whichis of the form Xlxdx, X' be- 
ing an algebraic function of «. The integration of this will alto- 
gether depend upon the form of the function X’. 

If n be an improper fraction, it may be reduced to a proper one ; 
but the final integration must be effected by series. 

233. The following examples wiil illustrate more fully the me- 
thods of integrating exponential and logarithmic functions. 


Ex. 1. Let the "ec tiaria be anna 


Let a* =z; ..dr= a and a"* = 2": hence the integration is 
reduced to 
atdz dz 
fot Jute) wins ater 
which is integrated by seriés. : 
Ex. 2. Let the differential be x™(ix)"dz. 


Since fa"dz = ay we find sts (232), 
ONS ere © Ni) festa es 
7 7 eee 


Substituting for n successively n— 1, n~2, &e. we fing 
21 


942 q THE INTEGRAL CALCULUS. 


nm. (n— +) 


Saum(lx)"da == na ae (Iay"——F 5 (le (La: cares Gn 1) 


(lz \n—2 . 
pert -(—2) | 
(m1) 


This series is finite when 7 is a positive integer: thus, ifn = 1, 
== 2, = 3, we find - 


— (lx a, &e. r 


ie ) , f 
“ Bo — I ba ke | 
: ere ee 


This is Puke to an exception when m = —1. 
Ex. 3. Let the differential be ue - ~~“ dx, n being a positive whole 


number. 


la)n 
Since = = dix; .. Oa = (lx)"diz ; and consequently, 


fae = 


; x n+1 
This is subject to an exception when n =—1. See ex. 5m 
Ex. 4. Let the differential be Ee, : 
Let z =a: samdz teat ¥ 
+1 ne 
; a! 
Also lz = (m+1) la: eee ¥ aa a 


Let lz =y; “. z=, and a a eMdy. Hence = 


amd eddy 
lx | 4) nom 


This is integrated by a series at 
Ex. 5. Let the differential be 2 (la ae n being @ positive integer. 
By (232), we find M ‘“ 4 


THE INTEGRAL CALCULUS. 943 


adr a9 : gmt smth 
ste - Te ~ (n=1) (lz)"—? Road Yaa 


By successively substituting rls n—2, &c. for n, we find 


ade anit § (In)—"  (m+-1) (In)" (m+ 1)%le) 5 
S as (ixy" ! I—n (i=n) (2—n), (1—n)(2—=")3—n) 
\ 
a Resi” i at 
% 6/1 Me a Se LANG 2 a). eed oie 
when m = — 1, the formula of reduction becomes 
dats l 
Saye =~ a (a= (n= 1) (lx)— —1 "hla abana 


This is liable to an exception, since it becomes infinite when 
n = 1. - The integral in this case is, however, easily obtained ; 
for, let z = Iz ; ; the, 


dz. dz ans sie . 
ee ee ee: 


and consequently » 


The final integral on 1 which if ae (ls - depends, appears, by the 
series just found, to be j in general f=“. Ex. 4. 


Ex. 6. Let the differential be /xdx. 
Integrating by parts “ 


‘ * Sladx = xrla—e = x(lx—1). 


Ex. 7. Let Py be the differential. 
ala? 


dz 


22 
ae Btu 1 
"Jaleo? zi 


dx — 


Spat era pis Uh 
Saye ai Mes va 1—x 


d . dx 
- x cla? 


ite 9. 


(244 THE INTEGRAL CALCULUS. 


SECTION V. 


The Integration of Differentials whose coefficients are Circular 
Functions of the variable. 


; % 
234. The elementary inteprals on which the integration of cir- 
cular functions, depends, besides those of algebraic functions, are 


derived from the following differentials : 
i 


‘ ; bai n sin np 
d.sinnd? = 2 cosnodo, d.sec ng = ad 
p no Q, | Q cos'ng Ps 
: ndo 
d.cosno =~ nodo, d.cotng =——— 
np ” Sin nodg, cot no Sinn 
nao | n cos np 
d. tan xp = ———, d. cosecng = — do. 
cos’np sin’ngy 
From which are derived the following formule : 
f cos Ronclilie sinnd sin ng do _. Sec np a 
n cos*np | n 
cos nO dh ~ cot nO 
sin no dg =~ ——— LS A) ee a Se 
Ss ait n Pee n° 
dy tan no cosnopd@ = cosec np 
cos nD on yf ~ sin’ng gst ne val 


235. When the arc or angle enters the differential Gincfiictent, 
itis generally disengaged from it by integration by parts, either i im- 
mediately or by the continual reduction of the exponent. The 
following formula will illustrate this. ~ 


[Xodx = ofXdx —fdofXdx ; 


where X represents: any algebraic function of x, and x represents 
any trigonometrical function of the are 9. Since d? must be an al- 
*gebraic function of x, if be considered as a function of 2, it 
follows that the integral fdg/Xdz comes under algebraic functions, 
and may accordingly be obtained by the rules alreddy established. 
Taking, for example, - fardx are A sin = x); if we at first inte- 


s 


. 


by 


THE INTEGRAL CALCULUS. ) 245 
&£ 
ll Ee “ P és dx 
grate x"dx, and observe that d. arc (sin = 2) = ———--., we 
shall find hyonits deat 
all fin pany , : d 
. é Po etl Be 1 atid 
and Cc = = — (sin = ee eS 
Surdx ar ing z) Pe | are. (sin = x) ial <A (1 — ah 
; ntl 
the formula [= ‘ ei has con ‘already considered in articles 


(212) and feist 
«286. When the differential coefficient is a function of trigono- 
matical lines only, the integration may be effected by various con- 
trivances derived either from algebraic transformations or from tri- 
gonometrical formule. ‘The. following are the principal methods. 
I. All functions whatever of trigonometrical lines may be re- 
duced to functions of the sine and cosine. By these means the 
proposed formula may be transformed from a circular function toa 
differential of another kind. Let the proposed differential, when 
its coefficient has been reduced to a function of the sine or cosine, 
be Xd@, X being a function of the sine orcosine. If x = sin ?; 
a dx 
he yc aay cane -. dg = — ay 
in either case the differential will, by the substitution thus suggest- 
ed, be reduced to the form Xdz, X being a function of x, and may 
be integrated by the rules already established. 
II. When powers of the sine or cosine of an arc occur in the 
differential coefficient, they may be developed in a series of the 
simple dimensions of the sines or cosines of multiples of that arc 


by the following well-known trigonometrical series : , 
ee ee sins adhe ooh sin nx — — sin (n~2) a+ 
7” ; Qu—t rr l “a 
ba ny i Ml (n—-1) * 
aby « ret oink WK: dices (n—4)x~, &e. ¢ ; 


this applies to the case where nis odd, and the upper or lower ™ 


is odd or even. 


sign is to be used according as . 


The series when » is even is 
= ~ cost 2 
= For ) 00s ne —7 COMM ya+ 


vie 1. 


sin"z 


4 T'\ 
a cos (n—4)a—, &e. f, 


| # The reader is referred'to Legendre’s Geometry and Trigonometry ; Brewster’s 
translation of this work has been lately published in New- York. 


\ 


i 


df 


246 THE INTEGRAL CALCULUS. 


and—or-+is used as > is, oda or even. ” 


III. When the sines ane cosines are connected by multiplication 
in the differential coefficient, tyeyemay be disengaged bv the for- 
mule 

2 sin z cos y = sin (x-+y)-+sin (x—y), 
2 sin y cos & = sin (x+y) —sin (x—y), 


2 cos x cos y= cos (x-+-y)+cos (x—y), ie ga 


2sinxsiny = Cos (x— te (x+y). 


Iv. Funetions ard the sine and cosine may always be converted 
into exponential functions by the formule 


2 cos x = eV (1) Leavy (1) 
24/(—1) sn z= C20 (1) oe GEV >), 
which may be established thus ; by (47.), Ex. 3, if 
u = cos 12:,4/(—1) sin a, 


du = =£4/(—1)udrz ; 


Be a t/(- 1) dx, 
and consequently 


“= tev) const. ay é 


But the constant riage be = 0, since, whenz = 0,u=13.., 
cos & ~£,/(-1) sn 2 = e=eV (—N), si ttn 


These two formule being added and subtracted, give the above 
mentioned results ; and therefore their integration may be reduced 
to that of exponential functions. 

V. When the differential is of the form 


sin™p cos"¢ d@, . ae 


it may be immediately reducedto a binomial differential, and inte- 
grated as in (208) by Byitng * 
sin @ = 2, cos g dg = dx, 
ss n—1 


. sin? cos" dd = (12) 2 dz. 
ol 


Or, by immediate integration by parts, the three following pairs 


“THEANTEGRAL CALCULUS. 247 


of formulz: may be readily obtained. In the first pair, one of the 
exponents is continually increased, while the other is diminished. 
In the second, one of the exponents is diminished, while the other 
is stationary ; and in the _third, one is increased, while the third 
is stationary. m 


sin™t!gcos"—!p 


m-+1 


f sin "9 cos odo = 


~~ ¥ > ‘a P , 
fsin™? cos a =— ear pcos"t ip | m— 


2 + fain+2gcos"—24d¢ 


1 
fain™-%ocos"**edo 


n-+1 n+ 1 
‘ . '¥ 
sin ™=!9 cos "Hp 
Jf sin ™p cos "dp Re rm =; /sin”2pc08 "odo 


— sin ™t! cos *—10 


J sin ™p cos *9d¢ = ome ¢ a if sin ™p cos "gdp 
m+! n+ 
J sin "$ cos "pdp = = ares aoe J aban ee 
inm +t cosnti ( 
f sin"pcos"odp = — fa = aaa a irae fsniieons ade | 


These formule are applicable, whatever be the values af the ex- 
ponents m andn. © 

237. These methods united with integration by parts, will, in 
most cases, effect the integration of trigonometrical differentials. 
Much of the facility of tlie. process must, however, depend on the 
expert 1ess and ingenuity of the analyst, which is only to be acquir- 
ed by E ractice; since no general methods can be assigned for ob- 
taining the integrals of these functions. The processes for inte- 


grating several general and useful formule are illustrated by the 
following examples. 


Ex. 1. Let the differential be 
1 de 
n J (1—2) , 


and sin nz=/ (12°) ; hence 


- da a 1 dz Ss, ty dz 
i ee a er er et ae ae 
* sin 27. ny |—~z2 n 2 — | 


dx 
—: let cos nt = 2; 
sin nx 


dz =— 


248 THE INTEGRAL CALCULUS.. 


. f= = +1} inners = i Wan a 4 
ee n " 


sin nx vt AS ata nz) ga 


Ex. 2. Let the differential £2 ee 
! COS NX 
In a similar way we find 


cos nx i 


fe aes 3! tan Me +n) =| tan (45°-+onz). 


xed 
Ex. 3. Let the differential be a = = cot ang: 


Since d sin x == cos xdx: yee 


Ex. 4. Let us takes the diferenval tan xdzx. 
Since sin xdzx =—d cosa; .. 


d d : 
po = f= =f er 


COS & COS & 


or ftan xdz =—I1 cos « = 1 sec a. 


d: 
Ex. 5. Let us take the differential —— 
tCcosz 


Since 2 sin x cos x = sin2x; ,, 
p 


| eae ae 
LE che A Bee = I tan 2, (ex. I.), 


sin x cosx sin 2x 


Ex. 6. Let us take the differential = 


—Ipdxz me 
Integrating by parts, we find 


sin 'adx _ die ladx 
f Py ey sin att fae 


Let @ = sin-'z; 


ladx 
| ies = fl sin ede, 


this may be integrated by a series. 
Ex. 7. Let the differential be x” sin ~!xdz. 
Integrating by parts, we shall find 


% 


THE INTEGRAL CALCULUS, 249 
eh oe Lo xt! gin sla 2tlde 
fx" sin dx = 


n4-L) tid Ate 


This integratign fails when n == —1. 
Substituting successively 0,1, 2, 3, &c. for.n, and replacing 
sin —'z and x by 9 and sin 9, we find 


1 
Jod sin @ = fp'cos dp = ¢ sin $+ 9 COS 9; 


‘a Yay ee 1 
/e sin 9 cos odo = 9? sin “o-hz sin ere ocr 


; VO) one ee ee’. 
S¢ sin 79 cos oda = 3@ sin me sin *@ cos 9+; 3 008 ds 


qin °.cos OPE go cos @ 


J¢ sin °$ cos dg = i? sin ‘ete 


Ex. 8. Let the differential be sin (m@--n) cos (p¢-+g)do: 
By method III. article (236), we have 
2 sin (mp-fn) cos (po-+g) = sin §(m-+-p)p+(n-+q) } 
+ sin {(m—p)p+(n—9)}. > 
Multiplying by do, and integrating by the formula (234), 
ip 2 008 }(m-+tp)ot(n+9) 
( do.=— 
Jsin (mert-1) cos (pera) ) Sin np) 
__ cos imap —P)gt(n—g} 
2(m—p)_ 


Ex. 9 ‘Let. the differential be sin “adx. 
Developing sin "x in a series, we find, when nis odd, 


1 ’ 
sin "¢ = MY “3 sin nem" sin (n—2)a-4- 


n.(m—1) .- if. ; 
ir. yh) eee 3 ; 


Multiplying by dz, and integrating, we obtain 
2K 


er 


te 


250 


3 . THE INTEGRAL, CALCULUS, 


© he a gaat | cos2z ncos(n— 2\a 
J sinnxdz = Be Sones cose p ) 


n Lo ne el 


n(a—1), cos (n—4)x ‘ ‘ 
1 NO 8), apna aims oe wie Me elke lene © 


In like manner, if n be even, 


4 “1 (sinnz nsin (n—2)x 
f sin ada = + ii 


2-00 n 1 nm2 tee 
n.(n—1) sin (n—4)a t 
Pe ns aera RS Sit ay, to Paes 
ee ; , in ¢ 
” By substituting successively for » the integers 1, 2, 3, &c. we 
find. ! 
f sin «dz = —cos «, 
ae ny slap Stash oa 
J sit adx = —_ sin ares 
lcos 3a 3 . 
f sin’ dx = Prien wae 
t 
Bea lsin4dz 1 
fsin'adz = grag ree sin Qos iy 


td e e e ° e e . 


Ex. 10. Let the differential be cos"xdz. 
Multiplying the series , 


| 7s) 
2"—Icos"x = cos nap cos 2) a+ A va 


cos(n Ayr, &c. 
by dx, mid integrating, the result is 
| sine | sin in (0 — 2) n(n—1) sin ont, 
2°—-Vfeos"xdx = ——— eae one. 8 
J ie ve ae « ee 


Hence, by substituting successively 1, 2, 3, Xe. for m, we find 


f cos dx = sin a, 


bE 
. fcos*adz = sin aes Ly 


a 1’ 


THE INTEGRAL. CALCULUS. 251 


% % 


. a 7 ‘ 1 ’* 3 
4 + a _ is 
cos*edz. =— sin 32-+— sin x 
J 12 4 ee? 


5 
&c: &e. 


f cos'adz = — sin fat sin 2a+s a 


Ex. 11. Let us taker the differential sin x cosnndz. 
Since d. cos. x = —sin xd, \ we have 
per cos”™™* ly 
Jfsin x cos"xdx =—fcos"ad . cos x = — —_.-. 
t \ ~ nt 1 
Ex. 12. Let ustake the differential cos xsin*xdx. This in 
like manner, gives 


. 


sin ait 
t¢ cos x sin*xdz = ——+— 
aa i 


' Ex. 13. Le ey cosnzdz be the diferentile 
Let cosa = 2; ., dz = — sin xdx, sina = (1 — 27) 


eal hha z"dz 
.. f sin’x cos"adxz =— 


v (l= 2) 


which has already been integrated, article 212. 


v x 


SECTION VI. 
Successive Integration. 


238. When a differential: coefficient is of an order superior to 
the first, as many successive integrations are necessary to arrive at 
the integral, or the primitive function from which it was derived, 
as there are units in the exponent of its order, and the same num- 
ber of arbitrary constants will be introduced. 

239. Let X be the differential coefficient of the second order of 


the function y, we shall ye e = Se and multiplying both sides 


“2 


252 ya THE INTEGRAL CALCULUS. 


3 ; ‘ . 2 
by dz, there will result oy = Xdz; now a is the differential of _ 
Hy 


d ths 
oa taken upon the supposition that dx is constant: we shall have 
xr 


sibrefore = = /Xdx. If P represents that primitive function of x 
which is equal to .(Xdx, and C the arbitrary constant, there will 
arise # = P+; multiplying the two members of this equation 
by dx, we shall find dy = Pdx-++Cdz, and by integrating we shall 


get y= /{Pdx+Cz+C’, C' denoting a second arbitrary constant. 
If. we replace P by /Xdz, there will result 


y =/dafXdz+Cr+C, 
an expression which indicates two successive integrations. 
We may reduce this expression to two simple integrals, by the 


method of integration by parts; for restoring P in the ager of 
fXdx, we shall have 


f[Pdx = == Px fel P = afd [Weds 
and consequently ¥ 
y = af Xdx—f{[Xadx+Cae+C. 
240. We now proceed to differentials of the third order. Let XY 
be the differential coefficient of she function y, relative to this . 
dy 


order : ve shall have ie == X, from whence 


ay 
a De = Xdx Sod e eaetae Oe. ; therefore 
dx? iis, 


oN 


Lak Peay 
qu = [kde $C, 


which gives 
d2y 


2 a = dxfXdr-+-Cdr. 
( 


Integrating again, there ho result 
ii, ote OY 


or from what we have remarked aide: 


THE INTEGRAL CALCULUS. 253 


uit = 2fXdeofXxde-+-Cr+C.. 


From this we deduce 


dy = tdx [Xda — dx [Xxdx-+-Cxdx+C'dz,,, 
and by integrating we get 3 ) 


C" being the constant introduced by this last integration. It is 
easily seen that 


fada [Xdz = : a*f{Xdx — : [Xx?dx, 


Jaa fXude = 2fX adx— f Xa*da ; 


substituting these values, and collecting all the similar terms, we 
shall find 


y =} (v/Xade—2efKadte[Xa4da)-+) (C#-+2C2C") 


241. We will now show in whatever manner successive integrals 
are denoted : when XY designates the differential coefficient of the 
second order, we have d’y = Xdz?, and by taking the integral of 
each member we find dy = /{Xdz*, and now integrating a second 
time, there arises 


y = f[Xdx? = frXde’. 


e 


We have in the same manner, when XY denotes the differential co- 
efficient of the third order, 


By = ide, dy = f[Xdz*, y = f{f{fXdx ='frXdz’, 
and so on for the higher orders. 


242. Hach differentiation introducing but one power of dz, we 
may leave this power only under the different signs f, which will 
furnish us with the following relations : 


[Xdx? = dxf Xdz, [[Xdx? = fdxfXdzx, . . 

i . | 
[Xda = dx?[Xda, f{[Xdx? = fdx'fXdx = dzfdxfXds, 
SjffXdu? = fdxfdxfXdx, Ke. 


where it is necessary to observe, that each sign / embraces all 
those which follow it. 


¥ 
a, 


¥ 


254. r THE INTEGRAL CALOULUS. 
ay. 
we 


This being Peomida, by neglecting the arbitrary constant, and 
integrating by resolution into parts, as above, we shall find 


f?Xdx? = <= aflde fers, | | Q 


pXdve= 40am 2x x dx+/Xade), | 


Sf Xdz* =—— tsps 303/Xrdx +-82f Kida —fXa%dx), 


&ec. , 


The numerical coefficients of these expressions are the same as 
those of the powers of the binomial a—6 ;. and whilst the expo- 
nent of 2 without the signs «diminishes by unity in each term, in 


proceeding { from the left to the right, its exponent under that sign : 


increases bythe same quantity ; according to this law, we shall 
have : 


SvXdan pore ee Ce Sax" 


n—1 } 
apne vray 
1a 2 a) ee) ee aS 


ees 2) napa Lon 3) fade 


ajar”. aa. 


The sign -+ of the last term being relative to the _case where 1 is 
even, and the sign — to that where itis odd. 
We shall restore the arbitrary constants which are : omitted in this 


formula, by writing 
[Xdx-+-C for Xda, fXxdx+C' for [Xxdz, 
[Rede OC" for {Xx Adi, 
ial so on for the others : for the constants, C, C', 0’, &e. being mul- 


tiplied into the different powers | of x, admit not sr further redue- 


tion. 
243. We shall conclude this section by observing, that the se- 


ries of Maclaurin (50) and that of Taylor (53), give likewise two 


general developments of the integral [Xdz. If we designate ahs Cc 
the value of this integral when « = 0, and represent by A, A, A" 


dX d 
&c. the values of X, oe &c. under these circumstances, we 
i 2 
shall have / ehh, é 


at 


: ‘ ng 
‘ THE INTEGRAL CALCULUS. 255 
ftae = CHA +4; as A+ at, &e. 
: ° . 
a.series in which C occupies the place of the arbitrary constant. 
If we commence with the greatest value of {Xdz, which is.re- 
presented by y, to arrive at. that which corresponds tox = 0, and 
which is denoted by C, it is evident that we must make s=—z, in 
the formula (53), which will give 


% 


Hiss By «x? dy 28 
C= —— ot, &e. 
Ide da? 1.2 de? ino awe 


iE ee : dy d24 dey d3y : 
substituting in this equation fe for y, aan hie &e. their values, 


and disengaging [Xdz, we shall have 
O° yh A ee a 8 . 
LO ge as Sees Bo” 
C denoting, in this case also, the arbitrary constant. 

244. The process of integration also conducts us to this deve- 
lopment : for if we divide the differential Xdx into the two factors 


X and dz, and.integrate the second, we shall have {Xdx = Xz— 
JfxdX 3; but | 


~ 


dX ED eel 
frax af Sade = 12 fx x 


da ar A tga ty Cackig Py, 
js eis 

: Xk -fex eee i JH 5x ‘A x fee 
‘de. . 3. de wo adas dx?” 
: OX = &X de no hs aX 

“da « dx" ms Ta a " dx? 

Ars 
&e. at 


putting successively for fedX, fx? jem he their values, there wili 
‘ ‘ ; ¢ ax ser 


result 


a dX. 22 PX . 23 

LAE Pee. da Vege | Oe 
And in order that the expression for the integral may be complete, 
we must add an arbitrary constant to this development, by which 
means it will become similar to the preceding. This series was 
first given by Joun Bennounti, whose name it bears ; and it has 


t 


‘ 


* + *& 
dye 


256 . THE INTEGRAL CALCULUS. 


> 


the same relation to the Integral Calculus, which that of Tay.or 
has to the Differential. 
245. By performing similar operations on fXardz, we shall find 


x Ls ay dX onl? HX ont3 
PAE XRT de WEY) de EIN FACTS 
aa abt Re 


GEIGED EEO) ee + Comet 


This done, since /* 2Xdx? = afXdx—f{Xxdx, we shall have 
(> ppp GX ee | ead © ot 


be ga ane ae 
[rXxda? = ‘ 
oC | ya? a¥ «9 @X at 
UA ae ean atl ae sr ylaal 


| gM ER Das’ PY dat 
+ Ae ede Beedle 2 Bk “2 xy ae 
we would arrive thus at 


x dX 3x4 | 2X 6x5 


3Xd 

LESS de aS 4 ie 8 St ee 
x4 daX 4x5 wx 1029 

Vdd! = C 

M3 2T3.4 de 2.3.4.6 det '2.3.4.5.6 7 


The coefficients of the numerators of. mee expressions are the 
1 

ee (Ifa) ae 

we ought to have, therefore, for the coefficients of the numerators of 

the development of, es the same as those of the development of 


same as those of the developments of ——- 


———— ; consequently, 


1+" 
ra i n(nt]) nse 
fied dX net On ey ee | 
Skee => List ee Ve, eae 1.2..(n+2) ’ 


&c. 


In order to have this development complete, we must add the terms 
affected with the arbitrary constants, according to the cy indicated 
in article (242), 


fei. 


* 


the parabolas of different orders, represe 
, Y 


: ‘ ; iP. 
y .* , 
w ®e - 
@ ¢ 
‘ > 7 “ P 
¢. . . 
o . 
THE INTEGRAL CALGULUS. 957 
w 


SECTION VII. 


* mn te 
Aven : 
The Quadrature of Curves. si 
6 é 


246. If an area a be bounded by a plane curve related to rec- 
tangular co-ordinates, its differential is expressed thus, (106) 
da = ydx 3. a = fydz. 


The equation of the curve will determine yinterms of x, and. 
the integral which determines the area assumes the form /Ydz, 


where X represents that function of, which is equal to the or- 


dinate y of the proposed curve: this integral miayebe obtained 


within any proposed limits by the methods already established. _ 

If the curve be related to polar co-ordinates, the area usually 
ebtained is that included between two radii vectores. Its differ- 
ential is expressed thus; (164), 


d as ‘ 2 e le nee 1 2 
‘a Fg wdt; * a = 5 Swede. 
By the equation of the curve, u? may be determined in terms 
of ¢, and the integration may be effected by the established me- 
thods. ‘The determination of the area of any surface is called 


quadrature, 2 
247. The curves which have the most oN integration are 


ted by the equation 


im 


4 — — Mi i 
y" = px™, from this we get y = px, and consequently 
- : r i . ‘ 
“np -—— 


“ 
1 m™ 

fidx = fp™ x" e= —\—x * + const.© 7 + 
All these curves, as we see, are guadrable, that is to say, we 
have a finite algebraical expression for the surface of the segment 
comprised by their arc, the axis of the abscissa, and the ordinate. 
The curves proposed pass. through the origin of the abscissa, 
since we have at the same time «= 0 andy = 0; if we wish to 
express their area, commenting from this point, we must suppress 

the arbitrary constant, since the expression ; 

2L 


» 


¥ 


Pal is eal to the area ae 7 and we c shal thus get 


= .» é 
te 7 
‘ “ 
* z ” : 
258 THE INTEGRAL CALCULUS. 
tS 
i) i 
1 
n +n *y 
np ONT ‘ 8 
m+n 


vanishes when we make aw = 0. To Ditnin the vali of the en 
BCMP, (Fig. 34), comprehended between the ‘ordinates BC and 
MP, which correspond to the abseissi® AB = a and AP = z, it 
will be sufficient to subtract from 


¥ 
7 


Wal + if 
emp’ Ps 
m—En F 
When the exponent n is even, the expression - 
se 1 ad € 
_np™ Mee 
pram | ” 


18 vanscagtltsl of the double sign +, +, and si since in that case 6 the same 
abscissee 1P belong to two branches of the curves ACM and 
Acm, we have two segments ACMP and AcmP ; that which, com- 
prises the positive ordinates has a positive value, and the other 
has a negative value. si 
if men 

When the exponent m and » are both odd, i quantity « * has 
but one sign, and continues always positive, whatever be the value 
of «; but itis easily seen, that in this case one of the two-branches 
of thelpr posed curve has its abscisse@ and ordinates negative at 
the same time ; it follows therefore from. this, that the areas cor- 
responding to negative ordinates and abscissz ought to be consli- 
dered as positive. ay tay . 


e . “ 
la 4g 


If n only is odd, then a quantity z ™ becomes négative at the 
same time with x; but in this case the two branches of the propos- 
ed curve are on the same side of the proposed ag si and the 
ordinates continue always positive. 7 

From these remarks we may conclude, that the area of a curve 


ae 
a 


% 


vowed = * 


% iy ‘ 
a 


» a 
THE INTEGRAL CALCULUS. 259 


us positive, when the abscissa and ordinate have the sume sign, and 
negative when their signs are different. 
_All parabolic segments have a constant ratio to the rectangle 


* ADMP, constructed upon! the wage a and the ordinate’; for the 


expression — iy, 
i nt 1% 
as nv mv 
2 —_= TG *s ais 
m 3m pr m-+-n is suc 
. ° - oi ad 
which is equal to | 
1 om 
“———xy, since y = i / 
tS ; i Si 


ap 


When m = n, the ie parabola banchican a right line, since orn 


L » 
case we Rie y= pre ; the segment. aCMP beeenses the triangle 


AMP, whose area | 


Ne 


I 
7 the formal above given, is sequal to oxy 3 


which is also known fom: Elementary Geometry. 


By oe tes = 3 one m = 1,.we have the common saree, 


and we find sy for the value of the segment CMP. 


248. We will now proceed to find the value of this segment in - 


curves, represented by the equation «"y° = p. This equation is 
deducible from 2 7 = px™, by changing m into —m; we have y = 
1 _m 7 


pax n, and consequent? 


e! + 


The curves proposed are hyperbolas of different orders referred 
to their asymptotes, and are composed of several branches, such 
as MMP, (Fig. 35.), inser id within the angles which the asymp- 
totes form with each other. If we réckon the seame s from the 
origin 6f the abscisse, they will comprehend the indefi nite space 
which is included between the part CY of the curve and | its asymp- 
tote AY ; the value of this space is infinite, or finite, according as 
mis greater or less than x. In fact, to get the value of the space 
BCMP, taken from the abscissa 4B = a to the abscissa AP = 6, 
we must successively make x = a and « = 6 in the expression 
/_ — nem ’ d % 
eso fee - 

——x ~ ,and thom? subtract the first result from the second ; 
u—m 
shall get 


w 


6 ; < 
260 THE INTEGRAL CALCULUS. 


1 
BcMp = 2? (0 > a”) * 
tte mM 
If we now suppose a == 0 ; the point B will coincide with theipeibe 
A, and the space BCMP sil be changed into YAMP ; now the 


n—m " 
quantity a will be ap or nonand: according as we have m > 
or <a: in the first case 
4 n—m ~ 
, ip” 71 
APM =—F_(——b “), 
| m-—-n\0 
and in the second — = 
a 154i 
ee ky Kae 4 
ye n—m n—m ie 
n 
yap = 22 5 ne -0) a | 
nm age 


Supposing a to be of a determinate magnitude, and making 4 
infinite, we shall then get the indefinite space XBCU, which will 
be infinite if m be lessthan 1, or finite and equal to 


% . Pa 
if m be greater than 7. It follows from hence, that when 7 and 
m are unequal, one of the asymptotic spaces are finite, and the 
other infinite. The reason of this difference i is founded on the 
sreater or less rapidity with which Ae curve apa oaches to its aye 


tote ; and since ‘ 
. 4 > 
Mio 4 1 % ; 
n m © ~ 4 ort, 
4 p , d =. p & : 
Y —— 79, Ane TF ae pine 
a “eS in ~ / 
A ¥ z sa iP 


itis easily seen that when we have m Sn, y decreases much more 
rapidly than «, and that consequently the curve approaches much 
more rapidly to the asymptote upon which the abscissz" are taken, 
than to that which i is parallel to the ordinates, and vice versa. 


iL im 


By putting y in the place of p" wt 


A 


, in the expression 


f ‘iad 
1 PY *. 
n—m vik | 


5 will 
} 7 main ae i 
hae » vr p" ey * 
 e@ « 9 Api 


ru—mM 1-7. bi p a 


it will become — ~ 2 yy and the val 6 the area VAPMY will he 


» r 


THE INTEGRAL CALCULUS. 264 


ney 


T—1 


“-+-const. It might seem that the expression et ought to 
vanish when x = 0; but what we have just proved shows the ne- 
cessity of making no inference of this kind before we have ‘substi- 
tuted, in the place of y, its value in terms of x. 

249, When n = m = 1, we have xy = p; the curve in ques- 
tion is the common hyperbola, and is also equilateral if the angle 
of the co-ordinates be a right one. -The general expression for 
the area found in the preceding article, presents itself in this case 
under an infinite form, whatever be the value of x, and the dif- 


ferential of this expression being vee has “for its integral pla-- 


const. The asymptotic spaces are both infinite in this case ; for x 
becomes so both by the supposition of « = 0, and by that of x 
being equal to an infinite quantity. 

rat ge = «and UMP, (Fig. 36), one of the branches of the 
equilateral hyperbola, whose power is equal to a’, and AC its axis ; 
if we draw BC a perpendicular to the asymptote from the vertex 
Cy we shall have 4B =a, and since the area BCMP = a7lAP 


P 
—ClAB = a’l a if‘we assume 4B = unity, there will result, 


since 1 = 0, BCMP = 1AP. We shall have.in the same man- 
ner LAP’ = BCM'P’, lAP” = BCM’P’, &c. from which it follows, 
that ifthe abscisse AP, AP’,AP”, &c. are taken in geometrical 
progression, the: corresponding areas BCMP, BCM'P’, BCM’P’, 
&c. will be in arithmetical progression. 

" 250. The hyperbola which we have just considered, being equi- 
lateral, has only furnished us with Naperian logarithms; but by 
varying the angle of the asymptotes, and always taking AB = 1}, 
we may obtain an infinite number of other systems of logarithms : 
Let UMP. (Fig. 37,), be any hyperbola whatever’; drawing the 
ordinates PM P,M,, &c. parallel to the asymptote ‘AY. we shall 
be able to prove by a process of reasoning analogous to that in 
art.(106), that the parallelogram PMRP’ is the differential of BCMP. 
Now if we draw P’Q perpendicular to PJV, we shall find P'Q=PP’ 
sin P’PQ = PF’ sin XAY; representing by », the angle of the 
asymptotes, we shall have P'Q = dz sin #, and consequently 


: 1 
PMRP =y dx sin a. If we substitute for its value—,. there 
vii ; *, 
ce. J é ‘ 
will result — sin # for the differential of the area BCMP, and con- 


sequently BCMP = [x = LAP, taking sin » for the modulus. 
The modulus of common logarithms being .4342945, (55), will 


» 


m2 THE INTEGRAL CALCULUS. 


sin # = .4342945, from which it follows that the asymptotes of the 
hyperbola, whose areas are expressed by the common tabular loga- 
rithms, make with each other an angle of 25° 55’ 16” 19’, nearly. 

251.. By making’ AC = a, AP = 2, and PN = y, (Fig. 38.), 
the equation of the circle ANE willbe y? = 2ax—<’?; and the 
differential of its segment VP will have for its expression 
dx,/(2ax—z"), which is transformed into —du,/(a°—u’), by 


| al 
making x = a—u, and which is also reducible to du (a?—u?) *, 


or Be a » by the formula ( B) i in art. (210), and the complete | 


Jf (a? — ey’ 
integral of it is 


1 ; 1 “ a—a2y 
—5 (a —2)./(2an —2*\-+ 5 a? ate( cos *); 


when we substitute for u its value, a result which vanishes when 
x= 0. 
We easily recognise in the part 


“ (a—2x),/(2ax—z"*), 


the expression for the surface of the triangle PCN, and conse- 
quently find that 


1 me 
a a? are(cos = oat)s or. or si AC. are AN, 
a 


is the value of the sector ACN. 
By supposing « = 2a, in the expeaggion for ANP, it becomes 


1 1 
5 a? arc (cos =—1) = Z ae, designating by « the semicircum- 


ference of the circle whose radius is 1, and it then becomesthe 


expression of a semicircle : we shall have therefore for the whole 
1 | 2” 
circle a’r = <a. 2aq. J 
The besibopaatt of f« if fe./ (2ax ~x*), which may be readily found 
by means of the series in art. ert) gives approximate values of the 


area APN. ‘ 
252. The ordinate of the ellipse being (ax — a"), the elliptic 


) b 
segment 4MP will be equal to _ Jdx,/(2ax —2°) ; and, as it com- 


mences at the same time with the circular segment ANP 
= fdx,/(2ac—a*), by comparing these two expressions, we de- 
rive the following proportion 


vile 
2. f 


THE INTEGRAL CALCULUS. 263 
e bh *, , 
AMP : ANP 3: = fdaa/ ((2a%— 2°) > fdxs/ (2ar—zx"), 


er, which amounts to the same thing, 


{ 
ee 
— 

we 


: 6 
area, of ellipse : area of circle 3: 
a 
& 


whence we obtain 


; b b 
area of ellipse = =X area of circle = — ra? = zab, 
: a : 


and this last quantity evidently represents the area of a circle 
whose radius is equal to ,/(ab). 

253. The nyperbala referred to its major axis is expressed by 
the equation 


 - 


aie b2 2 
y= a? (2ax-ba ); 


from whence we conclude that 
b a 
AQR = 3 fo dx,/(202-+2"). 


This integral may be found by means of logarithms (200), or it 
may be expanded into a series ; but instead of stopping to calcu- 
late these results, we shall proceed to the consideration of elliptic 
and hyperbolic sectors, whose differential uae ~ are of very 
frequent occurence. 

254. Let 4Bab. (Fig. 38. )s be an ellipse, whose semiaxis major 
AC =a, and whose semiaxis minor BC == 6, making CP = 2, 
there results ”* : 

| Ps ; 
PM = y = Y Hom 


It is evident that the sector ~ 
‘ ACM = CMP-+AMP, 
and that r a o£ 
d. ACM = d.CMP+d. AMP 
hud Iba... 
CMP = 5 CPXPM = oo (a), fe 


xidx 


d. CMP = = ash da4/ (a? —2* *\— Vaaay 


we shall have 


204 THE INTEGRAL CALCULUS?" 
. ob m.. 
8 8 dye AMPH = Re ag 


ws 
ad 
. » 


‘The last of these differentials is affected with the sign— , because 
the area 4/MP decreases when x increases ; and they give 
a? 
ar ee oe 
2 a,/(a? — 2) 
If we make : = 1, the elliptic sector .4CM will be changed into 


the sector ACN, which belongs to the circle 4Eae described upon 
the axis major as a diameter: we shall have therefore 


a’dz 1 adx 
; i! sone Move Vol o2—x?), ot > Hees ~x)? 
a * 
but — NP e = ) being the idrentiat of the arc AN, we obtain 


the same result as from the Elements " Geometry, which is © 


acy =4 axAN = q SACK AN, 


and since the sectors 4CN and ACM have their common origin 
in the point .4, we may then conclude that the elliptic sector 
ACM =~ ACN = 1} BOXAN. 
% - 
255. In the PiPerboie XAx corresponding to the same axes as 
those of the ellipse ABab, and ys So oe therefore is 


ie 


Vive om 
the sector ACR = CQR —IQR, which gives 
d. ACR = d. CQRe= d.AQRs 


d since | . | 
and since i 


3 1 er 1 eke 
oon Fenle s a), 
: 


d. AQR =~ dey/(at— a), 


tar. 


1b a*dx 


d. ACK = Ba Ae ay | 


op Ba ala aa al 


»THE INTEGRAL CALCULUS. 265 


from whence we see that the differentials of the hyperbolic and 
elliptic sectors aré identical, with the exception of the signs of the 
quantities involved. 

256. The hyperbolic sector ACM, (Fig. 37.), is equal to the 
asymptotic space BCMP ; for 


ACM = BCMP+-ABC— AMP, 


and 


anh OR sin B. APXPMX sin B__ ayep. 


co a 


257. The preceding examples are sufficient to show in what 
manner the Integral Calculus is applicable to the theory of curves ; 
we shall not, however, conclude this section without giving some 
interesting examples on the subject of algebraical and transcenden-. 
tal curves. 

Ex. 1. To determine the area of the curve whose equation is 
oi — ax?-+-a?y? = 0. 

Here, the given equation is reduced to 


a ria meena 


which, when x = 0 and area = 0, becomes -“5 ~ ; this therefore 
(a®—22)2 a2 _(a@—2)? 
: » leaves 5 aor 


plete integral, or the true value of the area included by the curve 
and the coordinates x and y. When y = 9, or, which amounts to 
ihe same thing, when z = a, the area included by the whole curve 


for the com- 


subtracted from — 


9 


: : a 
and the line of the abscissz will be ai 


¥ 


Ex. 2. Let the curve, whose ‘an to be determined, be the Cis 
ed 


2 aba te 
soid of Diocles ; whose equation bes = 


Here, we have 
2d x2 dx aide x3 
77 at ‘ 
eh yy eirens 2 & at 
a 


x 


266 THE INTEGRAL CALCULUS. 
r\ —} 
expanding (:-2) *, by the binomial theorem, we have 
a 


/ 


pyle ais ex (eda i” anak te elds Solas g, )s 


Brees by integrating, we find 


% 25 we x2 
sd de ORE (-) (¢ he & ). 
Sydx == x Vv “)* a, a if ASR C. 
Ex. 3. Let it be required to determine the area of the curve, 
whose equation ts a’y—a7y—u’ = O. 
, as 
In which case, y being = et 


—, we have 
e2 


vidx . «dx 


an 
a 


ydz = ape oe &ec. ; 


cad 


whence 


Syda ao a+ a aE ae 


5a3 § Tas 
This integral may be found without an infinite series by means of 
logarithms ; for, since 


£ 


fae ~i. 


ts — 22 aae 
it follows that ® 
akdx 1 Sadx ax 
Looe ay! | AS ee l ). 
Wier — 7¢2 a” a2 Lig? a a’ xX ea 


Ex. 4. Let it be required to anh the area of the curve, 
whose equation is a*y?--a°y? = a’, 
Here, by reducing the given equation, we get 


lis 


‘ a? Rott ee 2 2 Y 
Y= Taipaay? o. fydz = Lee yaar ae (a $a) 24C. 


In order to determine the valtte of C,let x = 0; then 0 = a?Xla 
+C; ».C =—a’Xla ; and consequently 


Sydr = a'l§ (e-ba/(al-bat)? -—a’Xla = al (ee Cee 


Ex. 5. To determine the area of those spirals which are repre- 
sented by the equation u = at. 


bare 


cu 


THE INTEGRAL CALCULUS, 267 


Since this is a polar equation, the differential of the area will be 
"dt : 
= , (164.) ; put for wits value, and integrating, there will result 


——— --const. 
4n-+2 1 


but the constant may be suppressed if we reckon the areas from 


the line .40, (Fig. 39.), where t = 0, and consequently the area 
att) f. Mh 
ACM = . After one revolution of the radius vector, we 
4n+2 


shall have the space 


oa a?( Qm\2nt1 
ACMB = 


x being the semicircumference of the circle OV; when the radius 
vector returns to the position 4M, we shall have the space 


ee. a ea ON lent 
_ ACMBCM = nas 
and so on for other positions of 4M. 
° ms me 1 ROK 
In the spiral of Archimedes, a = aon = 1, and. ACM = ET 


T 
3 
In the hyperbolic spiral, where n =—1, we find 


a result which becomes, when ¢t = 27, equal to 


a 2 
* ACM = ~=-+const. 


The arc of this curve, which makes an infinite number of revolu- 
tions round the point .4, is infinite when ¢ = 0 ; we must therefore, 
in this case, find the area comprehended between the two distances 
corresponding tot = 6 andt = ¢; this area will be found to be 


ay Ald 
SG-:): 


Finally, in the logarithmic spiral, ¢ = lu, dt = i ‘and the dif. 


9 


"dt du. 4 
ferential > becoming ss gives ACM = vs This area is no- 


thing when u = 0, in which case ¢ is infinite; for this curve, as 
well as the preceding, makes an infinite number of revolutions 
round the pole .4. 


w 
4,8: 
i, 


vr 


268 THE INTEGRAL CALCULUS. 


SECTION VIII. 


F) 


The eat of Curve Lines. 


258. it s express the arc of a plane curve related to rectangular 
co-ordinates, the differential of the arc is, (105), 
ds. = J (dy-+d2"), 


By inverting this formula, we find an expression for the arc itself, 
se fyay-tda) =f Sar bax, 


By the equation of ‘the curve, the value of , may be obtained 


in terms of «x, and the formula to be integrated in order to obtain 
the arc, will assume the form Xdz, X being a function of « ; this 


being integrated between any proposed limits, will determine the 
corresponding arc of the curve 


The determination of the length of the arc of a curve, is called 
rectification. ei, 


259. If the curve be expressed by an equation related to polar 
co-ordinates, the radius vector being Raercaenteg by My and the va- 
ryiable angle by t, we have, (161), 


= fi/ (wdt?+-du), 


fe caf ; wip OE Cae, 


mee : 2 
The coefficient _ and u, being expressed as functions of {, 
this formula may be integrated by the established rules ; or if 


eg be expressed as a function of u, and dt as afunction of u and du, 


the formula assumes the form Udu, U being afunction of u ; and, 
accordingly, we can integrate by the rules already known. 
260. Io determine the arc of a parabolic curve represented by 


the equation y = px", n being any number either whole or frac- 
tional. 


By differentiating, we find 


THE INTEGRAL CALCULUS, 269 
dy = npx"—"'da, 


ve dy? eda? =(1-bn?p?a2—)) da’, 


.s= SUF n2p?x2O—Y2 dx?, 


This can be integrated in a finite form only when 2(n—~ 1) is equal 
to unity, or when it is a submultiple of unity or 2, (207). In 
other cases the integral may be expressed by a converging 
series. 


3 : 
1 es =e 2n—-2 = 1. In this case 


8 =1( Yee 


4 


9. 9a. 28 
=e) I+5 pf -- const. 


a“ 


The curve pyspomed will be determined by the equation y = px?, 


ory? = p*x*. and consequently is the semicubical parabola, which 
is the evolute of the common parabola. 


In order to determine the constant, we see from the equation of 


the curve, that if we reckon the arc fromthe point where x = 0, 


we shall have 
i 


and consequently 


*  2Tp* ath a pr) 1g. 


4 


PO 
2 


To determine the class of parabolas* which are rectifiable in 


The origin of this integral is evidently x = — 


4 . ] 
finite terms, let m be an integer, and let m = age D ; therefore 
Qn, 
142 bn al 
n= nahh Hence y = px 2” represents the required class in 
mm. 5 
2 


this case. If 2(n—1) be a submultiple of n, let m = any” 


270 - He INTEGRAL CALCULUS. 


om 


-3; y= pum, In ge- 


being aninteger. Hence n = 
8 S 2mn—] 


neral, therefore, the number n is a fraction, whose numerator ex- 
ceeds its denominator by unity. If the denominator exceeds the 


numerator by unity, the integration may be effected by changing # 
into y, and vice versa. 


By making successively m—2 = — 4 &c. there will result 
5 7 
taco &c. which shows that the parabolas represented by the 


equations yt = p'z®, y° = p° x’, &c. are rectifiable: the réctifica- 
tion of all other araholas can only be effected by approximation. 
For the common parabola, in which n = 2, we have fda(1--4p? 


42)? : by the formula e of.art. (210), we find © 


a v1 d. 
fdx (1-+-4p?x?)? = 5 | -+-4p?z") we iqeee ray : 


° v N 
and since : neeT 


ie oe ! ya MEE 45 


ann de d . Sa Vile i ee 
Neyer) A pF ae AL Eds eh RN | 


there | result 


Se bare 5 altar ge i spoty Hhipeyt46 


This is the value of any are of the common par fabola we may 


suppress the constant in thie expression, if we SUP RARE the ental 
to commence when x = 0. 


261. To determine the arc of an hyperbolic, curve senkonestal by 
Cire 
The equation being differentiated, gives 


dy =—npx"—'—-dg, 
* ioe 
J (dy fda?) = (1--nip'e-—9)# dz, 


ie ds= = etn"? nt 2) de, 


and consequently, gg 


(aes fr (224 n2p?) 2dr, 


THE INTEGRAL CALCULUS. 271 


This does not come under the criterions of integration established 


in art. (207), (208), and can therefore only be obtained by approx- 
imation. 


adz 
V (ai aty° 


(105), according as we aBnle? the odiptiotis 


62. The differential of the arc of a circle is —— 
ada 

Pes / (2ax— 2?) 
y? = a*—x, or y? = 2ax—2?; neither of these expressions ad- 
mits of integration except by approximation. 

263. To determine the arc of an ellipse, whose equation rs a°y? 
“--b7a7 = 076". 

_Let this be differentiated, and we shall have 


: ben 


dy = — —-dx ; 
5 atypia? 
dx’. 
2. dy?--dz" “ay ao? 
But a*y’+-bia? = a°b?(a? ~ ex”), and ay? = a7b?(a?—2"), where ¢ 


represents the eccentricity. Hence 


a =f Zie ln Fe 
im of (a? —p } 
The series which gives the approximate value of this integral is 


‘given, when e is very small, in article 223, and which will apply to 
all ellipses of es ak eccentricity. 


If « = a = 1, the series for the quadrant of the ellipse becomes 


BO 


I 
[oo ee a ad 2G, 6 


x Did gy Sat eal vei ie: is 3.3.5, 
4 ‘ 4 aS 
a series which converges very rapidly when e is very small. 

%64. The differential of an elliptic arc is expressed ina very sim- 
ple manner, by means of the arc corresponding to it in a circle de- 
scribed on the axis major. of the .ellipse as a diameter. Let EN = 4, 
(Fig. 38.), we shall have 


CP = » =sin ¢ Cio 2 a 


v(l=2 *) 
and consequently 
| De de HEAR a Ae! Cn ah sith 1s 
265. The equation of the hyperbola being 


272 THE INTEGRAL CALCULUS. 


we have 
sai JK a? b?)\ 2? — atts 
Ta a) 
for the differential of its arc ;. making a = J, a?-+b? = 140? = 


77? 
this are will be expressed by fe i) cee, 
nearly equal to unity, be devatohed in a series by a process analo- 
gous to that in art. (223). 
266. It now only remains to make a few remarks concerning 
the rectification of transcendental curves. ‘The equation of the 


cycloid being 


——_, and may, when e is 


dam Che = 


dy /Quy—y'y’ 


we thence deduce 


dy,/2a 
J/(2a—y)’ . 


Vv (da? dy!) = 


a differential whose. integral is 


= —24/ § 2a(2a —y)}-+ const. 


But it is evident that ,/$ 2a(2a— y)} is the expression for the 
chord mg, Fig. (32), of the generating circle ; and as the variable 
part of the integral vanishes at the point K where y = 2a, it follows 
consequently that it expresses the arc MK: we have therefore 
MK = 2mg, AK = 29g, and therefore AM = AK —~ MK 
== 2(gq—mg): these results agree with that in article, (170). 

267. To give an example of the application of the formula: 
o/(wdt?-+-du?), which expresses the differential of the arc ofa 
curve referred to polar co-ordinates, we shall take the case. of the 
spirals which are represented by the equation u = at", and we shall 


have to integrate the differential 


B« ; 


dt,/ Salles Laie a at"—' dt (2-n2)2. 


When» = 1, we hae simply adt (+ 1), a differential of the. 
same form as that for the are of the common parabola (259) ; from 
which it follows that it is the rectification of this curve upon which 
depends that of the spiral of Archimedes. 


ru 


s 


THE INTEGRAL CALCULUS. 2738 


7 


In the logarithmic spiral, we’ have ¢ =u, which gives 
v (wdt+du’) = du,/2; the arc of this curve has therefore for 
its expression us/2-+-const., or simply u,/2, commencing from the 
origin of the radius vectors; and we see, that though there is be- 
{ween this origin and any point of the curve at an infinite distance 
from it, an infinite number of revolutions, yet they include an arc 


of finite length, which is equal to the diagonal of the square des- 
cribed on the radius vector. 


SECTION Ix. 


The Cubature of Solids terminated by Curve Surfaces of Re- 
volution, and the Quadrature of their Surfaces. 


268. A surface generated by the revolution of a plane curve 
round any line if its own plane as an axis, is called a surface of 
revolution. The quadrature of such surfaces is effected with 
greater facility than other curved surfaces, since they require but 
one integration, and are expressed by the equations of their gene- 
ratices. ‘or 

Let y = f(r) be the equation of the generatrix of a surface of 
revolution, the axis of x being assumed as the axis of revolution, 
and the co-ordinates being, rectangular. By the manner in which the 
surface is produced, it is evident that a section of it, by a plane 
perpendicular to the axis of x, and at apy distance x from the origin, 
is acircle, the radius of which is y. The circumference of this 
circle is therefore 2ry. If two such sections be imagined inter- 
secting the arc ds of the generatrix, the area of the circular zone 
or band of the surface between them will obviously .be 2ryds. 
This is therefore the differential of the surface; and if a be the 
area intercepted between two sections limited by any two values 
of x, we shall have 


as Qafyds, 
and by putting for ds its value, (105), we shall have finally 
a = Qafyy/(datdedy?), : 
2N 


* 


O74 THE INTEGRAL CALCULUS. 


the integration of which may be effected by ihe established me- 
thods. 

269. The process by which the volume included by any given 
surface or surfaces is determined, is called cubature. 

If the solid be one of revolution round theaxis of 2, it is evident 
that ry’ will be the area of the section perpendicular to the axis of «. 
If this area be considered as the base of a lamina intercepted be- 
tween two planes, the distance between which is dr, the volume ef 
this lamina is rfy*dx. When therefore we have the equation of the 
generatrix between y and x, we may substitute for y.its value in 
terms of x, and the integration may be effected. between any pro- 
posed limits by the established methods. 

270. To determine the surface and volume of a Mr 

A cylinder is produced by the revolution of a rectangle round 
one of its sides. Hence, in the formula 


a = 2/yda, 


y is constant; .°, a = 2sys, y being the radius of the base and s its 
altitude. Hence the surface of a cylinder is found by multiplying 
its altitude into the circumference of its base. 

For the volume, which is expressed by 


afydx, we have vy*x. 


The volume is therefore found by multiplying the altitude vf the 
area of the base. 

271. To determine the surface and volume of a right. cone. 

A right cone is a surface produced by the Jeligeatle of a recti- 
linear bode round one of its sides.. ‘ < 

The vertex of the angle being assumed as origin, and the axis 
of rotation as axis of a, the equation of the generatrix is ¥ = pz, 
p being the tangent of the semiangle « of the cone. ‘Hence, if a be 


its surface 
= Qn/yds ; 
but ds = y (de! + pa) = <= dara (\-tp?) § a 
4 n/p pada = ay (1p) px’, 


the eon of the integral being x = 0. 
Or, if s represent the side of the cone 


a= TYS. 


Since ry: is the semicircumference of the base, it appears that the 
surface of a right cone is equal to a triangle, whose altitude is equal 


& 
¥ 
bi INTEGRAL CALCULUS. 975 


to the side of the cone, ‘and whose base equals the circumference 
of the base of the cone. 

If the cone be. truncated, the integral must be taken between the 
limits x and 2’, corresponding to the distances of its bases from the 
vertex. Hence | 


a = /(I+p*). p(a?— 2) ; 
but (1—z’) v (i+p* = 3, the side of the truncated cone. Hence 


, =a. p(rpa')s = r(y+y')s. 


Hence the surface of a truncated cone is equal to a trapezium, 
whose altitude is equal to the side, and whose parallel bases are 
equal to the circumferences of its bases ; or it is equal to the sum 
of the surfaces of two cones, whose sides are equal to that of the 
truncated cone, and whose bases are equal to the two bases. 

To find the volume which is expressed by 


afyeda = = a[paida, 


we have, by integration, 


y 


+ afydx = ap" ie? =e sp gd 


the origin of the integral being e == O. 
Hence it appears. that the volume of a right cone is found by 
multiplying its altitude x by one third of its base xy, and that it is 
therefore one-third of a cylinder in the same base and altitude. 
If the cone be truncated, u representing its volume, we have 
1= ape") = gree) (atta! +2), 


Since x—2’ is the altitude of the truncated cone, let it be 4; and 
putting for the expression p(x? paca! rue); its value ey ry +y?, 
we shall have finally ‘ 


1 , , 
= gay ry ty"). 4 


ai 
1 1 
The terms — 5 my = 3 TAy” y?, are evidently the volumes of cones 


on the bases of the Need truncated cone, in the same altitude. 
1 
And the term = 3 tAyy! is the volume of acone in the same altitude, 


and having a balks which is a mean proportional between the bases 


? 


276 THE INTEGRAL CALCULUS. 


of the truncated cone. The given truncated cone is therefore 
equal to the sum of the volumes of these three cones. 

272. To determine the surface of a sphere. 

A circle being supposed to revolve on one of its diameters, 
generates a sphere. Let the equation of the generatrix be 


ye +e=r 
Differentiating, we find 
a dy 
dy =-——— 
eto: 7 
; 2 2 2d a? 
ve dy da? = pee dx? = - = 
y 7 


dx: 
*, ds = re and consequently 


a = Qafyds = 2afrdz = 2ar(xu—~x), 
the origin of the integral being «’. 


If x =r, the formula expresses the volume of a spherical seg- 
ment, whose base is 4 lesser circle of the sphere at the distance x 
from the centre. Let that part of the axzs of the segment (the 
diameter of the sphere passing through the centre of its base) in- 
tercepted within it be called V, and we have 


a = 2erl. 


It is evident that 2rV is equal to the square of the chord C of the 
arc, whose revolution generates the segment. Hence, 


a= nC, 


The surface of the segment is therefore equal to the area of the 
circle described with this chord as radius. Hence the surface of 
an hemisphere is equal to the area of a circle described with a ra- 
dius equal to the side of the square inscribed in a great circle, and 
the entire surface of the sphere is equal ‘to the area of four great 
circles, or to the area of a circle described with the diameter of the 
sphere as radius. | * 


The formula * 
fe a@ == 2er(c— x’) 


_ expresses the surface of acylinder, of which the altitude is (r—2’), 
_ and the radius r, (270). Hpaice it appears, that if a cylinder be 


THE INTEGRAL CALCULUS. 274 


circumscribed round a sphere, so that it will touch the sphere both 
with its sides and bases, the part of the cylindrical surface, inter- 
cepted between any two planes perpendicular to its axis. is equal to 
the part of the spherical surface intercepted by the same planes, and 
the whole surface of the sphere is equal to the cylindrical surface, 
exclusive of the bases of the cylinder. The spherical surface bears 
to the entire cylindrical surface, including the bases, the ratio 
24.8: ins 

If round the circle, whose revolution generates the sphere, an 
equilateral triangle be circumscribed, one of. its vertices being on 
the axis of revolution, it will generate a cone, called an equilateral 
cone, from the circumstance of the diameter of its base being 
equal to its side. It appears from plane geometry, that the alti- 
tude of this cone will be 3r, the radius of itsbase 4/(3).r, and 
therefore its side 24/(3).7. The conical surface of this cone is, 
therefore, 67r’, or equal to six times a great circle ; and since its 
base is 37’, its whole surface is nine times a great circle. Since 
the circumscribed cylinder, including its bases, is six times. a great 
circle, the three surfaces of the sphere, cylinder, and cone, are in 
geometrical progression, and in the ratio 2: 3. 

273. To determine the volume of a sphere. 

The formula 

u = afydz, 


becomes by substituting for y? its value r?— x, 
| 1 
usaf(r—sz*)de; wu = a eae ©); 
the origin of the integral being « = 0; or 
ae, , 
t= ws r(x ) Re (x2°— x”) i 
; G2 J a0 ! 19 
w= a(e—2') 2 (2*-+2x'+a yg, 


the origin being z = 2’. 
To determine the volume of a spherical segment, let « = r; 


i 
= at(r+2) § D7? ae raz}. 
To extend the integral to the entire sphere, let 2’ —=—r; ., 


aed 3 
ee 3 


which is the volume of a sphere whose radius is 7. 


278 THE INTEGRAL CALCULUS. 


Let a be the surface of the sphere. By the last proposition 
a = 4zr’: hence, . 


u == -ar 
3 b] 


which is the volume of a cone whose altitude is r, and whose base 
isa. Hence the volume of a sphere is equal to that of a cone, 
having its base equal to the surface, and its altitude equal to the 
radius. 

The volume of the circumscribed cylinder i is 2rr°?; since 2r is 
its altitude and rr? its base. Alsothe volume of the circumscrib- 
ed cone is 3zr’, since its altitude is 3r, and the radius of its base 
./(3) . r. Hence it appears that the volumes of the sphere, cy- 
linder, and cone, as well as their surfaces, are in geometrical pro- 
gression, and in the ratio 2: 3. 

This beautiful property was the discovery of Archimedes, who 
was so charmed with it, that he is said to have ordered it tobe en- 
graved upon his tomb. 

274. To determine the surface and volume of a prolate spheroid ; 
which as the body that the ellipse generates by revolving round its 
major axis: the equations of the ellipse referred to its centre being 


lias ne a », 
by substituting this value of y? in the formula ry°dz, we shall have 
? | 
acydx = sory x*\dx ; 
and by integrating, we shall find 
b? 3 
frpdx = 4 («=-5) +C. 
suppose the integral = 0, where x =—a, we shall hgve 
go", 2 
es Cringe Ss 
nd by substituting this value of Cin the above equation, it becomes 
R 1 ae i ti : oe 
; " me a x3 2 
it Bo at hy. Dike de 3 ae 4 
faydc = iS (a aa +30 ) 
Th order to comprehend the entire body, or, which amounts to the 


same thing, to have the definite integral, it would be necessary to 
take the integral from x =--a to a = --a ; and we should obtain, 


THE INTEGRAL CALCULUS. 279 


6? 4 (4eab? 
frypdxe = sai i oe 


the expression for the volume of a prolate spheroid. 
And we should also have 


2b x2 
~ afd a*— (ers 


for the expression for the area of its surface. Let a?—6? = c?, 
and this expression will become — 


2 
fied FSned 


now, it is evident that this integral is that of the surface of a circu- 
g 3 . i a 
lar segment, whose abscissa is « and whose radius is nr 


When a = 8, the proposed body becomes a sphere, and the ex- 


: : 4ra3 
pression for its volume becomes 


, the same that is found by 


the Elements of Geometry. a 
We should also have, when a = b, 


Mei 
» 


f2radz = Moan const. 


for the expression for the area of the surface of a sphere, which 
gives, by taking it from x =—a to x = -+a, 4a? for the value of 
the whole surface of the sphere. 


SECTION X. 


The Integration of Differential Equations of the first Order 
and first Degree. 


275. In all that we have hitherto said on the subject of the In- 
tegral Calculus we have supposed that the differential coefficients 
were expressed directly by means of the variable on which their 
primitive function depended ; but most commonly we have merely 
a differential equation in which these different quantities are in- 


280 THE INTEGRAL CALCULUS. 


volved. Forthe first order, the differential equation, when it is of 


the first degree with respect to dz and dy, has necessarily the form 


 Mdu+-Ndy = 0, 


‘ 


and it expresses, as we have already shown, the relation which 
subsists between the variable 2, the function y, and its differential 


coefficient rh 
dx ; 

One of the most simple methods of discovering the primitive 
equation from which this derives its origin, when it can be effected, 
is the separation of the variables, or the reduction of the equation 
Mdz+Ndy = 0 tothe form Xdx-++Ydy = 0, X being a function 
of Xalone, and Ya function of y. In whichstate itis immediately 
integrable by the rules for integrating differentials of a single va- 
rlable, the integral being 


fXdu-+-fYdy = 0*. 


276. In differential equations of this kind, the variables are said 
to be separated, and therefore all equations in which such a sepa- 
ration can be effected, may be considered as integrable by the above 
method. The most remarkable classes of equations, in which 
this can be effected, are the following :, 

I. All differential equations coming under the form 


Ady+ Ydx = 0. 
{I. All differential equations of the form. 
XYdy+X'V'dx = 0. 


IIT. All homogeneous equations ; that is, all algebraic equations 
in which the, sum of the dimensions of x and y in every term is 
the same, and which, therefore, come under the form 


Kea, cindy, 2 ays oi ca hy (am cer, ay? 1.) RO. 


~ 


* This would be, according to the usual custom, expressed f/Xdxa4/Ydy = C, 
C'being an arbitrary constant. This, however, as has been very properly observed 
by Larpner, is superfluous, since the introduction of the constant isa part of the 
operation indicated by the sign /. The constant should therefore be generally 
neglected, except where the integration has been actually effected; then it is proper 
and necessary to introduce it, because the symbol which implies its introduction 
has disappeared. 


THE INTEGRAL CALCULUS. 281 


LV. Linear equations ; that is, equations which involve y and dy 
only in the simple gaat at and. which therefore come under the 
form 

dy+(Xy+-X') dx = 0. 
V. The equation of Riccati (an Italian mathematician), 
dy-+- (Ay?-+- Ba) dz = 0, 


in which, in certain cases, the variables may be separated. 

There are other equations in which the variables may be sepa- 
rated, but these will sufficiently illustrate the principle. It is evi- 
dent that all equations which, by any transformation, may be reduc- 
edto any of the preceding forms, may be integrated in the same 
manner. 

277. The equation 


BY 


* 


Ady Ydz = 0, 
being divided by XY, is reduced to 
dy | dx Fh 
the integral of which is immediately obtained, 
dy dg 
; Lx thee? 


If, for example, the proposed equation was «dy-+yda = 0, the se- 
paration is immediately effected; for, dividing by xy, we should 
find | 


aa dx 
—t4"" = 0, 
y 4 
and by integrating, it gives nates 
ly-Ple = Cy oplay=C: | ang 


%& 
since we may consider the arbitrary constant as a logarithm, we 
may therefore assume /xy =Jc. Passing from logarithms to num- 


bers, there would result zy = c. 
278. The equation . 


# XYdy+-X Vide = 


being Giaied by X Y’, becomes 


yt sda = 0, 


S 0 


mr 


282 £HE INTEGRAL CALCULUS, 


which is immediately integrable, 


[pat gle = 0. 


This process is applicable to the differential equation 
a*yda-+-(3y-+1)dy,/a? = 0: 
for, if we divide by ay we obtain 


ihe eg oe -dy = 0, 


ye 


the integral of which may be ay found by the established me- 


thods. 


279. Each term of an homogeneous equation being of the form 
Ay"x"—", the constant sum of the exponents being m; if every 
term of the equation be divided by x", the form of each term will 


become a(t). if 2 - = 2, the equation will assume the form 


J eddy + iz) de = 


But since y = az, .°. dy = xdz +-2zdx ; which being substituted 


for y, gives 


af(z)dz+ 3 F (2) pale) jae =0;° 


mee: tae oy Ba ; 

“Fete te 

or Fe) Setee = 0,0 Le 
f(z) 


which is of the form 
: | Bde Xda = 0, # 


in which the variables are separated. — a 
Let us take, for example, the equation : 


a*dy = yda-+-aydz : 
eee iw 
by making y = 2x, we ofp have This % 
ag = zdr+axdz, | sibs 


oh and, by substituting these values, the equation will become 


vadu-ade = 2°e*dr-+-zaxdz ; 


* homogeneity, ohagette dx and dy into dz‘ and dy’. is) 


THE INTEGRAL page. 283 


reducing and dividing by x°z’, we will obtain 
. 


a 


and by integrating, we shall haye 
Peat em Lis 
py BUn f y 
Let us take, for a second example, the equation 
(2°+-yr dy = (w@—y)ydz : 
by making y = za, and reducing, this equation will become 
(aha. 3) ae 
dx I+-z 


: = 


putting for = its value derived from the equation y = zx, we shall 


have angie 
ade ZC Lane ™m 
jl ‘‘ 
whenee, by transposition and reduction, we shall find 
ade wa T-Lz 
" = aw Oe8 dz; % 
and finally aati = 
| = (2% af=s sink = = =p +3 ae 3) | 
22S Qz. «Bz ¥ 


280. Equations are frequently rendered homogeneous by substi- 
tuting for x and y, «+a and y-+d, and disposing — of, the arbitrary 
quantities a and 6, so as to take out the terms which -destroy the “Ga ” 


For instance, the equation | Te PA 
(a--mz--ny) tO retards = 0. am 
may et am made homogeneous by subalOHinh i ae inthe 


place o ay +i in that of y shall have die = dx’, dy=dy', 
and | ; 
SR, hte: Pen’ +ny) da’ +(B Fpatgbtpe’ + gy')dy = 0. . 
: i 
We may make the constant terms disappear, by assuming ir 4 
© ° 
: a , * 
a 4 ‘ 


ee tt, Be be er ok Ole De pa Oe eae Se ey ae ee ee eee Ae 


> 


284 * THEMANTEGRAL CALCULUS, 


e-+matnb ="0, altho, = we 


from which equations we may determine the quantities a and 6; 
and there then results the differential equation © 
ani 


(ma 4- ny’) da’ +-( pa! +qy) dy = 0,. 
which is homogeneous with respect to the new variate 2 x and ¥. 
The preceding | transformation is the same as that which is made 


use of, in order to change the origin of the co-ordinates upon a 
plane; it gives no result when mq—np = /0,a case in which 


a and & become infinite ; but then we have g = eit, and  conse- 
7 m 


quently 


wa px-tqy = L (ma--ny) 
Latta sy " 
the proposedequation being changed into " 


4 


ads : ee een) (ae+?. iy) =yk 


it is sufficient to Piro mat ag = 2, \in order to my ait the va- 


riables. } * 2k. 


Substituting this value, as well as that of dy, which results from 
it, and disengaging dx,we find rapes F oo. 
— jee (8m-- pz) er . Y a * 
“ emn — pm?-+(mn—pm)z Pe C4 : 


the integral of this equation will involve logarithms except in the 
case where mn—pm = 0, when it will be ¥ 


e a Pa Ses « 2eme+pe? 


fe 4 2(amm — Bn?) am 

d i dd ie 

The transf mation etnias ed in, this last case has changed the 
equation ‘into another which — , 


i itis readily seen, that. wh the bquoyee which we have 
thus treated, we shall be able ge it the f da+Zdz = 0, Z 


? 


ie 


j being a function of z only, “a 7 _ that we may thence deduce 


& at/Z 


nie 


281. In ate equation J 


dyt(Xy+X')de=0. 9 


iy Let Xz = y; .. ‘3 X"dz-+-2dX", by which. substitutions 


the proposed Moriah becomes 


> 
» 
ri 


iy 


eS / Stay [\-* t AE a 


~ 


Ne 
m7 5) 
on 


THE INTEGRAL i ae 
ie, X'dz+2 AX" XX" zd -bX da = 0, 


in which X” is an arbitrary function of x; weareat liberty there- 
‘fore to assume it ia such a manner that the preceding equation may 
“4 be divided into two others in which the variables may be separated. | 


Now it is easily seen that this condition will be satisfied if we 
make 


Adz XX"2zdz = 0; «. 2dX"+X'dx = 0. 
If we divide the first of these equations by X”, it becomes 
dz-+Xzdxu == 0;. 


from which we deduce 


mh 
7 : 
<+Xdx.= 0; .*. U4 
¢ Zz 4 . 
fit ; a oe € 
and passing from logarithms to numbers | 
> —- e—/Xde - e mY 
tr: we neglect here the arbitrary constant, as it will be sufficient to 
| add it at the end of the operation. Afterwards deducing the value 
» of dX” from the second equation, and substituting in it the value 
of z, which we have just found, we shall have © at 
AX" = mel X'dues *, X" =—fe/XaryX’ dr, 
' ee h : 
and consequently i” © 
é y = —e SXdefe fXde X’ da, 
Let us take, for example, the equation 
| dy (y—axt)de = 0. ik i ae . 
In this case, X¥ = 1, AX” ==-—- az: .. pe te 
fXdx = x, and ie o ele. oa 
ere 4 bi: 
ee ‘But “ bern * 
be a aide = = aer(a° Bx" +64 —6) ; is 
hence the integral sought i pe % a 
i, 2, “a ie a 
» a g— Ca pax? Sat Gare "i a a 
In the equation of Riccati, 4 Wot? 
ge. or 
i ne r dy+ (Ay?-+ Baum) dx = 03 oe 
v7 % ad igi” . - 
ifm = 0, it becomes : | | ag 
=O, f “. * 
% : % 
1 : “ s % 


miter 


—s 


2986 THE INTEGRAL CALCULUS. 


in which the variables are separated, 
But if m be vot = 0, let 


By which substitutions the given equation becomes 
ade +-Aztde+ Bamt4de = 0. 


If,in this equation, m == — 2, itis homogeneous ; and if m==— 4, 
the variables may be immediately separated by dividing the whole 
equation by «7(Az?+-B). 

If, however, m benot =~ 2, nor = —4, a further transformation 
must be effected. Let ' 


z=-,a™3=>= u; 
‘and let 
m4 A —B B A Pil. 
i = = == i a ea 
m4+-3 ~ m+3’ m+3 


We find by these substitutions, that the equation becomes 


dt,/(A'?4-B'u")du = 0; ‘ 
this being similar to the first oy can be integrated when 
m =—2, orn =—4. . 

If n be not = ~2, or n = —4, ‘by continually repeating the same 
transformation, the equation may successively be reduced toa 
series of equations of the same form as the given one, and in 
which the exponent of the variable becomes successively equal to 


m+4 n+4 p+4 
~ on-3? n+3’ pt+3’ 
mt4 — 3m+8 — 5m412 ™m-+-16 
m3” — 2in-5" 3m+7’ 4m+9 tie 


The equation can ‘only be integrated by the methods above given, 
when some one of these is either = 0, =—2, or = 7 ea that 1 is 
whenm is a number coming under the formula | 


—An a. Sie, een d 
Sey being any positive integer, or = 0. 
T— 


THE INTEGRAL CALCULUS. * 287 


; 1 ; , ' 
if the transformation y = Tr gmt! = z had been made in the 


given equation, the same process would show that the integration 


—4 i BE 
could be effected. when m = saa; The criterion of the inte- 


grability of the given equation by this method is then 


—4n 
Qn)’ 


m= 


n being a positive integer, or = 0. 
Let us take, for example, the equation 


dy+ (y2—a2e 5) dz = 0. 


” 4 . -_ 
In this case the exponent 3 comes under the character 


4n 
2n-+ 1’ 


since they agree whenn = 1. Let 


% 


ap w= t-3,y = 3a7%2—' 5 
me *, dz-+-(2?7—9a7t-4) dt = 0, 
»and consequently, by integration, we shall find 


ebae’ } y(1++- Ban") +30 ; a 


y (bm 3ax*) 30% 

& 

283. It is proper to observe that an equation which has been 
obtained by the immediate process of differentiation, 1s called an 
exact differential ; it is the same with respect to any differential 
function whatever, which has been obtained by means of differ- 
entiation only. When a differential equation, such as Mdz-- 
Ndy = 0, is not an exact differential, we cannot perform the pro- 
cess of integration until we render it an exact differential by some 
preparatory operation. 

284. Evrzr was the first who resolved the following important 
problem : 

Ist. A differential equation being given, to determine how we 
may know whether it is an exact di ifferential ? 

'¢ 2dly. What is the means of integrating an equation whichis not 
an enact differential. ; %, 

Before we proceed to the solution of this problem, it must be 
recollected, that a differential equation is not always the immediate 
“Scat 


* 


e 


288 . THE INTEGRAL CALCULUS. 


result of the differentiation of a function of two variables ; 
but that it most commonly arises from the elimination of an arbi- 
trary constant, between the primitive equation from which . every ty 
its origin, and the immediate differential of this equation. — 

285. The elimination is effected immediately, when the: primitive 
equation is under the form u = c, w representing any function 
whatever of x and y ; for, by differentiating, we have du = 0; if 
the function du has no factor by which it can be divided, it will 
always preserve the form of a complete differential of two va- 
riables, and the equation 

7 
dy du 
BS Cea 4 
. dzdy» dydz Aa 
furnishes the means of discovering this form ; for when we have 
du = Mdx+ Nay, | 


there results from it 


du 4, ut. 4 a dM. dN | 
dx \*ydy@ ? drdy. < dy @ dx” « mm 
and consequently» _ ve 
dM dN | 
dy dx ‘. 


it is necessary Mereuame that every faction Max-+-Nay, which is 
a complete differential, should satisfy thisequation ; and when this 
condition is fulfilled, it will be easy to ascend to its integral, since 
then we shall have = 


a 
a 


hy Ae? du 
io dae idly: 


which gives us the values of the partial differentials. 


If we take the value of the partial differential, relative to 2, for 


example, there will result 
du 


and consequently ) 
u = fMdz+Y. é (1). 


We add in this case an arbitrary function of y, since the inte- 
gration has only taken place with respect to one of the variables ; ; 


THE INTEGRAL CALCULUS. Pa 289 
2, 
butare this function may be decgitinass since the function of 
S ea 
¥ isfy the equation V = ape * ) %: 


vation “ — ape Ea? m g 4 


9, “ du _ dfMdx , dv ee 
« ye dy dy’ 
Be JMdz By », we have P ; . 


7 dus dy my N 
, ay ~ dy dy iy 


trom whence we dedifte e. 
: | : | 
4 dY dv 
g =: — = N- — 
ae * dy dy 
—. a hres 
and by integrating ight ay 


ee one ee 


we shall find therefore 


ion. Pace tien Pe het 


which is the integral of the fon propagedy ? 


This result shows that the funetion N - ~ ought to involve the 


‘s > ¥ | 
variable y’ ‘only, otherwise it would not be true, as we - supposed; 


that Mdx and Ndy were the partial perce of the same funce- 
tion u ; and by developing this roe we are conducted to the 
equation of condition” 


aM dN 

me dy fr de’ 

* 
which was before found by considerations of an inverse nature. 

mee Itis evident, that in order to obtain 


1 3 dy dy b] ‘ are ee 


2P 


we must | ite y-tdy for y, in the ation {Mdzx, which will 


290 . THE INTEGRAL CALCULUS. 
Si (ae “rt &ec. ) te, = TH eel Tes ded, &ec. 


2: (Mdx-+dy fe a a, &e. 
since the sine / 1s relatibe to the variable x only ; we shall have 
therefore , ¥ 


YMde dM f.. 


dy 


substituting this value of 


‘ ry ah 
¥ 


there will result 


y= fi Gi Aes de)dy, (0 


By taking the differentials, at first with respect,to y, we shall find 


ae dM Ny 
aii yf if 
and afterwards differentiating with reference to x, we shall finally 
get ’ ie 
dx * dy 


286. By Meus of the f gral (1), we may integrate any func- 
tion of two variables which satisfy the condition of integrability. 
Let us take, for example, the function Pe 


ees ja = 7 or pls Site aaj 
By comparing this expression with the formula Mdz+Ndy, we have 
consequently the conditiot of integrability is fulfilled, since we find 
sat _ 22? —y? dN 


~ @ yy da? 


integrating Mdax, we om. 


». - 
THE INTEGRAL CALCULUS. 291 
_ pyda ‘ia es 
j JMdx = iceapat = 72 = are (‘an = =) 5 « 


whence, equation (1) gives 
Ae 


- 1. are (tan == ae 


al 


4% 


Differentiating and considering the whole as variable, we shall 
find < 


__ yda—axdy 
ay? 


comparing this with the proposed function, we shall have 


du —-+dY; 


dY = 0, whence Y = const. 


and consequently 7 


de—ad . 
fat ae Y — arc (tan = ;) tent: 
Again, let us take the equation . 


dx ydx yd dx — «xd 2a) 
de de pened) 120 2 
ot ss ae x 2y 


By comparing it with the formula Mdz-+Ndy = 0, we have 


c wile 


*_ vw ety) iad SRT 


a i. a oR a 


2y° 
and we find x ‘ | 
. dM _ 2yt / (a®-+Ly?) y? 
8 Rl Sve 
e)  aN GytaVety) 
ae Mess le 2 a TY: 


i ; 
these values being equated and reduced, give 


dM aN 2y Myrgspiny? |) 


cot aeeten I 


‘dy . dz Pea (x8 ty)’ { , 


and consequently the proposed equation may be integrated imme- 
diately. We first obtain 


id 


292 p THE INTEGRAL CALCULUS. * 
: 


pia = we tf By Reta 
249?) 16 Pay 
bai f ~ 4) (x2-+-y yee i mat v(EA | ¢ 


mee AGS by!) 
Do2 


therefore, hin la — : 
ae : , 


. a 
1 § -ytJ/ (ety!) ey 
T3 ce ae ae ney 
We afterwards find F 
. wn — 
dv (x? 4-4") y Pais 14. 3 
IR TF TS OtR) B/E) |B By? 
and finally Y= sl from whence there results 


sa 4 


“u=Iix— 


a J (+y?) aby ty) 
a” on oa Ta flo }= ey <7 ry 


The form of this example was too complicated, i. make it pos- 
sible to recognize, by inspection alone, whether it was a complete _ 
differential or not ; and in all similar cases it will be necessary to 
commence by a Urigts nti whether the proposed prmaton satisfies 
the condition of integrability. 

-287. When the primitive equation is not of the form nu = ¢, OF 
when the differential du = 0 includes factors which afterwards dis- 
appear, tlie resulting differential equation of the first ‘order no lon- 
ger admits of fritattliate integration. If we had, for example, 
u = y—cx = 0, we should i ‘ 
du = eg hi 


and eliminating ¢, there would result cdy— "a = 0, an equatio 
waco does not satisfy the condition of integrability, since it gives 


i dM dN : 
= yy N= Xs dy Ses == jh. 
But if we Aisenkage the constant c, we shall have c = 7, and by 
‘% : 
differentiating, . 
r, We ng 
4 ae = 0; 
& 


under this form — 


“hh 
x 4 , 
THE INTEGRAL CALCULUS . 293 
e ‘ 
» 2% 
Be ircge goth haMy ey tae ae 
¢ ' a oa dy x8 dz’ 


we see therefore, that t the integrability of the sate” ady — yds 


1 
= 0 depends on the restitution of the famior — which has disap- 
hdd 


peared after the differentiation and the elimination of the. apibery, 
constant. e 

288. In general, every erential eauatita, in which dx and dy 
do not exceed the first degree, must have resulted from the elimi- 
nation of the constant c, in an equation of the form P+cQ=0, P 
and Q representing any functions whatever of « andy. We find 
by this elimination QdP — PdQ = 0, whilst by SE Raa a the 


P 
equation —- ek we should have obtained _ 


°Q. 


9 QP—PIQ _. 


i i @? 
*, * ; aad ; : 
rm - *, 


1 
the first method of pibceadine causes the factor -—-, to disappear ; ; 


Q’ 


and with it all the Rictors may disappear likewise whieh are com- 
mon to the two quantities QdP and PaQ. “ 
It follows from what we just observed, that when the equation 


Mdz-+-Ndy = 0, 


does not satisty the condition of integrability, it is because differ- 
entiation and subsequent elimination of the arbitrary constant con- 
tained in the primitive equation, have caused a factor to disappear, 
which, if it were known and restored, would render the first member 
of the proposed. equation a complete differential of two variables. 
Let z be. this factor; we shall consequently have 2Mdz-+-zNdy 
= %'; being the primitive function we x and 4, and therefore 


“* d. 2M __¢@- aN 
dy dx 


‘ ¥ 
By developing this last equation, we shall find 


dz dM dz , zd.N j 

M igh Upp. des Lav sh Minetecaense 
, ™ 
dz @dz ,sdM dN - 
——— ais act —_—_—_— = a7 0. ex.s ry 
% am dy dz ( dy dx )e (1) 
4 

# 


294 {HE INTEGRAL CALCULUS. 


ni , 

' Since VM and NV are, supposed to he Biven functions of « and y, this 
equation, when integrated and» solved for z, would determine its 
value. It being, however, an equatior of partial differentials, its 
solution can very seldom be effected ; and even when it can, it 
presents generally greater difficulties thas! the proposed equation, 
since the function z which it involves depends on two variables, and 
has two differential coefficients. 

289. Although we cannot therefore in general determine a factor 
which will render an equation integrable ; yet there are some pro- 
perties of these factors which merit attention. 

¥. If the integral of the differential 


2Mda--zNdy 


were known, the factor z could be found ; for the above formula is 
identical with 


du du . | 
) % * ae pasne dy, ¢. 
~~ « ok 
therefore, by comparing them, we shall obtain. | ~ 
' AK 
du 


Set ie a a 


the quotient is independent of the differentials dx and. | dy. 
a likewise obtain the factor z, by equating together the values 


off 7 ¥ deduced from the equations " ' 
da’. 
du ‘ 
ot de bar Fa = 0, Mda-}-Ndy = 0, 
which gives ~ a rf ihe » 
du . 
‘ ee 
du N’ 
dy 


and consequently shows that if M and V have no common factor, 
z will be the greatest commony divisor of the differential coefficients 


dus 
Midna oY bo 


THE iNTEGRAL CALCULUS. 295 


il. If we know a value of 2) we may deduce from it an infinite « 
number of others, by observing that if we multiply the two mem- 
bers of the equation zMdx-+-zNdy = du, by any function whatever 


of u, which we will vs ih by $(u), the two members of the re- 
sult 


2(u) Mdx-+z29(u)Nar = o(u)du, 


will be complete differentials ; that z being a factor proper to ren- 


der the equation Mdz-+Ndy = 0 integrable, ee 4 ies 20(u) 
will possess the same property. 


Ili. The factor z may, in some cases, be a faitetion of one of 
the variables only, and then it is easy to obtain an expression for it 
by means of the equation (1) in the preceding article. If, in this 
equation, we suppose - = 0, it will become 

dz dM _ an 
et da e Co dy =>) = 


from whigh we shall deduce 


d 1sdM aN 
ag ne od, 


If the second member of this equation be independent of y, 
then z is a function of x alone, and not otherwise. 

Since Mand WN are given functions, the value of z can always 
be determined by the equation 


1 {dM dN y 
= f—{ —d ek. ie — 
= fel G8 -a | ix) = fXdz ; 
% “ 
a 2 =e fxaz, "& 


We will not stop to consider the case where the factor need only 
involve the variable y ; we will readily see that the expression for 
it will then be e/¥¢¥, by making 


1ydN amy es 


and that this only applies when Y is absolutely independent iof a. 

290. Homogeneous functions have a remarkable property, which 
enables us to assign the ‘factor which renders an homogeneous 
equation integrable. ‘To explain this property, let «represent an 


4 


ge 


296 THE INDEGRAL CAL CUINS. 
_ homogeneous function of « and 4 In u, let # be changed into 
«(1--2) and y into y(1-+-2), and let the function become w’, so that 


u = f (2, Y)s uw = f(x-pix, y+tiy). 


Since u is an homogeneous function, u’ = (1--7)™u, m being the 
number of dimensions of the variables in each term of w, let these 
two values of uw’ be developed, the one in powers of ix and zy 
by (140), the other in powers of ¢ by the binomial theorem. 
Hence | . “e 


d2u, 7x7 “du iy ae Py? 


du 
vaate ssh dy! der. 8 4 dxdy’ jn, 8 dy? 71. “yg SE 


1 ar ee _ ge Spt edi cin — 2). hi Kc.) 
Hence, by P piatine the corresponding coefficients, we find 
du du 
mu == da ae? 


m(m—1) au x ahs ay, au y es 
eA 4 dia? lus 2" dady At * dy®¥1 . 


It is evident that this property belongs to homogeneous punches of 
any number of variables. , 


291. Let the equation to be integrated be 
, Mdx--Nay 4), 


where M and N represent homogeneous functions of the variables. 
Let z be the factor sought and ico an homogeneous function, and 
let it be supposed that » , 


is an exact differential. Hence 
¥ 


®. d( Mz’) __ d(Nz) 
dy da 


Let the dimensions of JM be p, and those of wu, 13 .°, 


aa) d(Mz) 
da 


(p-+n) Me = es 


a+ 


es | ae 


val 
THE Bi Wt si | 297 
Rayna Sat mart Ne), 


’ RPA = a 2 eee By: 


+ . 
and consequently * 


cp tae ryate fat) 


BS 


This equation is fulfilled by the conditions 


¥ = — (n--]), 


1 
| ~ MabNy” 
Hence the equation . 
* Md Nady _ ey Mi: 
“Ma+Ny , 


as integrable. 

292. The methods of integrating differential equations of the 
first order and first degree, shall be more fully illustrated by the 
following examples. 

Ex, 1. ./(1-+y?). dz—ady = =. Dividing by x./(1+y7°), we 
have ; 


. 


dx dy of 
SIFY) 
which is immediately integrable.* F . 


Ex. 2. Let (Ar+By) dy+(42+By) de = 0. Let y = zz, 
and dividing by 42-+ By ; .we haye 


but dy = zdx-+zdz. Hence the equation is reduced to the form 
Xdz+Zdz =. -* ct 


Ex. 3. ay"dy-+-(a2™-+-by) dx = 0. If y = zz, 
+ RA 


* In general, in examples we shall finewed no further than the reduction of the 


equation te.one which is integrable by a former rule. 


2Q 


“ts 


~ 


x 


998 THE INTEGRAL CALCULUS. 


da “azz 
we SE Be ees Pam 
RAR Toot foan 4 


* 
Ix. 4. ndy aydz = 4/(x*--y?).dz. Let y = 2a, ., 
dy—zdx == dx /(1+2%), 
pe ae 
* ahs: / (1-42?) 
3 
Ex. 5. Find the curve whose area = b.. Hence 


= 
[Ge 


4: Differentiating and multiplying by a? : ue have 


x*ydx = 3xy?dy — Pride = 0, 
or (x*y-+y* dx —3ry*dy = 0, th, 
which being homogeneous, let y = zz, 


, dx  3zdz oN 
SS nh UE BP uc’ 2 
- 


* . «(1 —227)3 = C, 
and consequently 
t | (a®°—2e?)? = Cx’, 
is the equation of the sought curve. 
Ex. 6. (1-+2*)dy—(ya-+-a) dx =0. Hence, 


i 
dy thal 


were dx = 


Here we have 


x a 
ee a A, Ge a 
1-+2?’ 1+22 


Hence, fXdx = —5 (1-0?) = —14/(1-+2%) ; also, 


d. 
fesXde X' dy: em f(a “Fs wie = [——, ; 


+72 


are ae ’ i i 
; ' ; ee 


4 


THE INTEGRAL CALCULUS. _ 299 
but . & ‘ 
Be ala C: 
(ay con | 
- Hence the sought mugel is 4 
Pi ene te 
Ex. 7. Let dy— ( rt) dx =0. Inthis case 
Ate las * X oirat ‘ 
% | h 
2, fXde = alla); 2. ees (l—2)*, 
j e—/Xde= se. 
(ae oe 
hence the sought int ny ’ 
ence the sought integra is wi 
1— # 
4 y= OTS Ae Sal 
. (1—2)* l-+a . 
Ex. 8. Let dy+(y?= 2") dz = 0; ; 
. = i 
ee da: a 
and consequently * e 
Ley 
eh bore a 
Fx. 9. Let dy-+(y?—aa—*) dx =0. Let 


ae: d } 
y a 2? t b} 


2 


a : eG ices re | 
\dy = 0. 
ithin. the criterion of 


Ex. 10. ape ae 
This equation will be found not to come wit 


x integrability, since 
dM dN _.  Bbyz 
“dy dx Jatey 


But since V = 2by/ (L-+a*) ; _ 


= ES 


300 THE INTEGRAL CALCULUS, 
# 
. WG ES) hi x 
N Pag? ?, 


which being a function of x alone (289) case 3d, the equation will 
become integrable if multiplied by a function of x... 'To determine 
this function, let it be z. By (289), case 3d, ge. 

% 


le ~ Sire =— gta 
1 
~ Otay 
Multiplying by this, we find 
Grae bo) d+ 2bydy = = 0, 
which is integrable. : 
Bx. 11. Let a®dy-¥$42°y— ata Barer Psi In Btls case 
ts BA eee uN Oe 
Syd fy * 
A a oo hy 
NG Wy ae ae 


This being a function of x alone, a factor may be determined which» 


_ will render the equation integrable. By (289), case 34, the factor is 


* 
." t, 
a dx 
ote ey 1, ss See a 2 eer. 


Hence the equation being maultiplied by x gives 


atdy+§ 42° yoo ~x) des 0, 


which ¢omes within the criterion, since ° 
dM dN _ 
dy dx 


b m 
Ex. 12. Le ae Cal \ fe dx 


| y . 
The multiplier is xy, and 


crm tarl 
oe CMY = ——-+(; 
Y m~-a-+1 Bi. , 
Bx. 13. Let cdy—2ydxtade=0. 
% ey 


THE INTEGRAL CALCULUS. » 301 
¢ 


. beat GeO ‘a 
The multiplier is mS and 


* a 
or cx? — = 0. 
x at 2 


SECTION Xt. 


The hat ibation of Differential Biteghivtts of the first Order; 
and which exceed the fet Degree. | 


‘et oz ‘ 
pri 


3 
293. It appears by the ordinary process of differentiation, that 
no differential equation, obtained directly by differentiating the pri- 
mitive equation, can exceed the first degree. But when between the 
primitive equation and its immediate differential a constant is elimi- 
nated, which enters these equations in any degree superior to the 
_ first, the result will be a differential equation of the same order as 
before, but of a superior degree. ¢ 
Every differential equation of the first order, whatever its degree 
may be, must be comprised in the formula 


i 
(By an(Bys0( ee... tulle’. 


Let the roots of this equation be p, p’, p”, &c. hence it may be 
expressed 


d dy, 
(Fr) Gr) Grr) +. vy =e 


‘This equation is resolved into the several equations 


dy— “ode — 0, 
dy—p'dx = 0, 
dy— pdx = 0, 

7 &c. 


T.et each of these be separately integrated, and the integrals be 


302 THE INTEGRAL CALCULUS. 


U =0,U' =0, U"=0, &c. Any one of these integrals, or 
any number of them combined by multiplication, will satisfy the pro- 
posed differential equation. For,, 
dU = dy—pdx = 03; d(UU') = UdU'+U'dU = 0; 

“. UU’) = U(dy—p'de) +U (dy—pdz) = 0. 
It is obvious that these conditions are satisfied, and that the same 
will apply to the product of any number of them. 

294. But a difficulty presents itself from the consideration that 
an arbitrary constant is introduced in each integration, and that 
therefore arbitrary constants are introduced in the integration of 
a differential equation of the first order, which seems contrary to 
the principles in the general theory of differential equations. 
This however is krcornued for thus: the constant, by the elimi- 
nation. of which the differential equation of the nth order was ob- 
tained, must have entered the primitive equation in the nth degree, 
and therefore it had different values derivable from that equation ; 
the n arbitrary constants, therefore, thus introduced, are. only 
these n different values of the constant eliminated. 

295. The n differential equations of the first degree, into which 
the proposed equation has been resolved, may also be accounted 
for by mere differentiation. Let the primitive equation be imagined! 
to be solved for the constant, of which, therefore, 2 values will be 
obtained. Upon differentiating the equation, each of these values | 
will give a distinct equation of the first order and first degree. 
These equations are no other than the simple factors of the differ- 
ential equation of the nth degree. 

296. The following two examples, although very simple, will 
remove all the difficulties which are connected with the preceding 
statement. : 

Ex. 1. Let dy’?— Saint = 0: this equation is decomposable into 
dy-+-adx = 0 and dy—adx = 0, whose integrals are,y--axr = ¢ 
and y—ax ==c’. We readily see that each of these results sa- 
tisfies the proposed equation. The equation (y-++-ax—c) 
(y—ax—c’) = 0 satisfies it likewise, for it gives 


(y-bax—c)(dy—adz)+(y—ax—c')(dy+-adx) = 0, 


fe 


from whence 


_ i (ytax—c) - — (y — axz—c’) fade | 
2y— (cc) 


. . e "t i 
putting successively, instead of y, its values c—az, car, we 
find 


3 


THE INTEGRAL CALCULUS. 303 
dy =—adz, dy =+adz. 


The mtegral (y4-ax—c)(y—ax—c’) = 0, involving two arbi- 
trary and irreducible constants, might appear more general than 
those of the other equations of the first degree, which only involve 
one constant ; but we must keep in mind that each of its factors 
ought to be considered separately, and that we deduce from it no 
other lines than those which would result from an integral including 
one constant only, of which this equation is likewise susceptible. 
This last integral is obtained by making dy = mdz in the differ- 
ential equation dy*—a’dx? = 0, which is thus changed into 
m?—a’ == 0, by which the quantity mis determined, whose value 
we ought afterwards to substitute in the integral of dy = mdz, 
which is y = ma-+e. It follows from this, that the integral of the 


proposed equation is the result of the elimination of m between 
the equations 


y = mz-+-c, m?—a? = 0; 


if we effect this operation, there will arise 


i bis Me 


This primitive equation being of the second degree, gives for each 
particular value of the constant, two straight lines inclined in dif- 


ferent directions with respect to the axis of x, which is also the 


whole that is furnished by the other integral (y+ar—c) 
(y—ax~c') = 0, excepting that each factor only represents lines 
inclined in the same direction ; but since by giving separately to c 
and call possible values, these quantities will necessarily pass 
through the same degrees of magnitude ; by collecting together 
those straight lines-which correspond to the same values of the 
constants ¢ and c’, we shall fall in with the solutions comprised inthe 


(ee) — a’ = 0, which is limited to the single constant c. 
x 


We ought to observe, that every equation which involves dy, dz, 
and constant quantities, may he integrated by making in it, as above, 
dy = mdz. 

Ex. 2. Let us now consider the equation dy’~—axdz* = 0; 
we deduce from it 


dy-+-dz.faz = 0, dy +dx. fax = 0, 
and by integrating we shall have 


304 THE INTEGRAL CALCULUS. 


% 


‘These equations, as well as their mre may be separately con- 
sidered as the integrals of the proposed equation ; “but this case is 
different from the preceding. since the radicals which the integrals 


just obtained involve, have with each other a connection which. . 


gives the means of comprehending them both in the same equa- 
tion, and with one constant only. In fact, if we make the radical 
disappear in the equation ee 

a4 /(ax*)\-—c = 0 
vt5v( x*)—e ual dh 


° 


4 f eS Abul ay 
we obtain (y—c)? = 5 ot: This result is)still the integral of the 


proposed equation, to which it will immediately conduct us by the 
elimination of c. It belongs toa species of parabolas, each of 
whose irrational equations represent but one branch; and the 
product of these equations will correspond to. groups ‘of branches 
belonging to different curves, but which, being collected together by 
two and two for the same values’ of the constants, would give 
nothing more than the rational integral. 

297. By what has been just explained, it appears that the inte- 
gration of differential equations of the first order and superior 
degrees depends on the resolution of algebraic equations. But as 
our powers in that department of ate, are extremely limited, 
several artifices have been suggested to elude the necessity of the 
resolution of the differential San ; we shall therefore explain 
the principal of these. 

If the differential equation conta only one of the two variables, 
for instance x, and the differential coefficient p, and can be resoly- 
ed for x, it will give 


a = F(p). 
Now, since dy = pdz, integrating by parts, we find 
y= pu —fadp = px—fF(p)dp. 
Thus, the integration of the equation is reduced to that of the for- 


mula F'(p)dp, which can be effected by the rules already established. 
Let us take, for example, the equation 


vda--ady = b,/(da?+-dy"), 


or a--ap = 6,/(1+p"), by writing p in the place of if 


- 
a 


THE INTEGRAL CALCULUS. : S06 


This last equation gives immediately 


/ 2 =—ap+b/(1 +p"), vg =—ap-b/(i+p*), | : 


and Conamaaently * 
by, eo o= bp +p r a fp 4p") 

298. If the proposed differential equation contain both variables; 
one y, entering it only in the first: crane then solving the equation 
for y, we find — bes A 

y = F(z, p) = r; 


i 


dV dV | 
. dy = “fe ne 


ay 


But dy —=pdns 0, 


dp diy rie 
te (5 —P) ar age #Q, 

If this equation can be integrated, an equation of the form 
f(z; P» c) = 0, 


will be obtained. © 
pe this and the proposed equation, » being eliminated, an equa- 
tion between x, y, and an arbitrary constant, will be the result, and 
will also be the integral sought, . 
Let, for instance, the proposed equation be 


al 


adie = wg! (da? ay?) F 7 
* 


‘hes 3 = Pp, and ‘wansposing we have byt i 
y= a a 
Bes: 2 : ee y 
But dy = pdz ; therefore | ' 
} dx a i t, 
ae Pye ap Tit 


~~ 4 
a 
yr 


and consequently 
ly = —14/ (1 tpl pr ¥ (itp?) } le ; 


wae Spot’) , 
2k : 


t= 


J 


306 THE INTRGRAL CALCULUS. 


and y= 5 Pv 4p % 


Eliminating p, we find 
ped 0; xfy?tQcx = 0: 


the former is contained in the latter, bee what it UbGomies when 
C=O. 

299. The following formula, Which’ is involved in the’ feinerat 
case is very remarkable, and of very extensive application, and its 
integration is very easily effected. 

If the proposed equation could be put under the form y= pet P, 
and if P involved the ‘coefficient p only, we should then have 


dy = pas+(2+ =) dp; and since dy, = pda, there will. remain 


the equation» uv 
7 we . 
(+=) dp = 0, wei 


which may be decomposed into the two factors te = 0, and 
dp = 0. Eliminating p between the first of these and the pro- 
posed equation, we shall get a primitive equation, which will» sa- 
tisfy the one proposed ; but which, involving no arbitrary constant, . 
will only be a particular solution. The second factor - being inte- 
grated, gives g =c, or dy =cdx, and y= cate ; The con- — 
stants c and c’ are not both arbitrary ; for, by making in _ the pro- 
posed equation, p = c, we have y = cx+C, C being what P 
becomes by this substitution, and from which we infer that c’ = C ; 
the integral of the proposed equation is therefore y = se ioe and 
is found by changing p into c. © 
Let us take for an Tl the equation . 


ydx ~—xdy = = = nf (da*+-ay'). 


eS ae Rf i 


It may be put at once under the form. we 
y= papi (EP 5 ; 
and i differentiating, we find , tie he 
d; 
dy = pdz-+-ad mtn 
ei ri 


and since dy = pde, there will remain 


i 


THE INTEGRAL CALCULUS: 307 


pie MTP Na 
v J (ip) 
This equation may be 5S bio? into two factors, 
med belle = 0, anddp = 0; 


_v (1-+p*) 


te rh ¥ 


the second facior leads to p = ¢, and the integral sought for is 


Yy == cab ite /(1-c?): 
The first factor gives 


» 3 : 
nS ni 


Substituting in. the proposed equation, we have y?-+a* = n?, an 
equation involving no pbitrary constant, and which is not included 
in the integral a + 


y= extn / (1 +c’), 


and which is nevertheless of such a kind, that the values of y and 
dy, which are deduced from it, satisfy the proposed differential 
equation, of which it consequently offers a particular solution. 
300.. It must be here observed, that two methods of deducing 
differential equations from their primitives or integrals have been 
explained in Section VII. of the Differential Calculus ; one by 
direct differentiation, and the other by eliminating a constant be- 
tween the primitive equation and its immediate differential. Let 


F(«,y, ¢)=0 be the primitive equation, c being the constant designed 


a 


; d ti 
for elimination, and let f’(x, y, c, p)=9, ( where p= ), be its 


immediate differential, obtained by differentiating the former for x 
andy. Eliminating ¢ by these two equations, let the result. be 
F(z, y,p) = 0. ‘This equation being independent of c, will evi- 
dently be the same whatever value be ascribed to c in the primitive 
equation. When c in the equation f(x, y, c) = Ois taken as an, 
indeterminate or arbitrary constant, this equation is called the 
complete integral of the differential equation F(x, y, p) = 0; but 
when a particular value is ascribed to c. it is called a wale m- 
tegral, as being only a case of the equation in its general state. 
301. It does not, however, necessarily follow that the complete 
integral, including an arbitrary constant, contains all the primitive 
equations from which the differential equation F(x, y, p) = 0 may 


308 THE INDEGRAL vaLcuLys. : 
be derived. It certainly include’ all the particular integrals, that 
is, all those which involve an arbitrary constant ; but there may 
be certain other primitive equations, which, containing no arbitrary 
constant, are not included under the formula f(z, y, c) = 9, and yet 
from which the equation F(x, y, p) = 0 may be deduced. Such 
equations are,therefore entitled to be considered.as integrals equal- 
ly with the equation f(x, y, c) = 0, Such, integrals * are called 
particular or singular solutions, as opposed to the integral f(x, 4, c) 
== 0, including the arbitrar ry constant, which is called the generai 


salution. 
302. Particular solutions,t of sinh the equation ydx— ady == 


ina/ (da?-+-dy?), (299), has afforded us an instance, is; intimately 
connected with the differential equation from which it originates, 
although it cannot be referred to any of the cases of the complete 
integral, whatever value we assign to the arbitrary constant, as it is 
éasy to pes by comparing the equations y = cx--n4/(1-+c’*) 
and 2*-+-y? = 

303. The se solutions, bvithout being explicitly compris- 
éd in the complete integral, may havevibeliias. be deduced from it, 
if we cease to look upon the arbitrary constant as invariable, In 
fact, let V = 0, be a primitive equation containing the variables 2, 
¥ and a constant c ; the corresponding differential equation, which 
we will designate by V == 0, will be the result of the elimination of 
this constant between the equations ie = 0 and Getta 0; 
but if we suppose ¢ to bé any function whatever of x and Yy, We 
shall give the equation U = 0 generality sufficient to represent any 


* There is a species of solutions which may satisfy a differential equation besides 
those which are considered in this section. Let Mdx+-+-Ndy —0 be a differential 
equation, and let any function of the variables, as f(a, y), be supposed to be a: commo® 
factor of Mand JV. It is obvious that f(z, y) =0 and Mdz-}-Ndy = U will be fal- 
filled at the same time. In this point of view f(z, y,) == 0 may be considered asa so- _ 
lution of Mdx-|--Ndy=0. Such solutions, however, are not comprised in the pre- 
sent investigation. ‘They may always be found by determining the common divisors 
of Mand NV. These solutions ought not to be termed integrals of the proposed equa- 
tion, becanse it does not follow, that being differentiated, their differentials would be 
equivalent to the proposed, which is the specific character of a primitive or integral. 

+ This species of solutions occasioned considerable embarrassment to the earlier 
analysts, and were held as a kind of analytica: paradoxes. Euler considered themas 
forming exceptions to the general rules of the calculus, and gave methods of distin- 
guishing them from ordinary integrals. Clairaut also determined a class of differen- 
tial equations which admit of singular solutions. The complete exposition of the theo- 
ry of singular solutions, of their connection with the complete or general solution, and 
‘of the ‘circumstance’ from which they derive their origin, has been given by LAGRANGE, 
m1774._ 


ss 
Tiip INTEGRAL CALCULUS. © ° 309 


given equation between the two variables, and consequently all the 


pageeenet solutions of V = 0. This being premised, throwing the 


dU dU 
equation or dx+- gt = 0 into the form dy = pda, we shall 


observe that since the equation V = 0 results from the elimination 
of ¢ between U = 0 and dy = pdx, it ought to remain unchanged, 
whatever value we assign to c, and consequently that we are at li- 
berty to suppose ¢ variable, provided that the law of its variation be 
such as to allow the equation dy = pdx to continue true ; now, al- 
though when c is regarded as variable as well as x, we Mee in ge- 
neral dy = pdx--gdc, p and q being functions of x and of ¢, still, 
provided qg = 0, we have dy = pdx : determining, then c, in_func- 
tions of x and y, by this equation, and substituting in U = 0, the 
the value which results, an equation will be obtained which will 
still satisfy the differential equation V = 0. 

In what has’ been said, y was regarded as a function of x 
and c, considering in its turn a, as a function of yand c, the 


aU : 
equation -_ dz+ my dy = 0, may be thrown into the form 
x 


dz = mdy ; and reasoning as before, we shall find that if the value 
of dx, taken on the supposition that c varies, be dz = mdy+-ndc, 

the equation resulting from the elimination of c between % = 0 and 
U = 0, will also satisfy the differential equation V = 0. 

304. The primitive equations afforded by both the foregoing pro- 
cesses are necessarily either particular solutions of V = 0, if it be 
susceptible of any, or particular cases of its complete integral. 

These two processes may be comprised in one, by getting rid of 
all fractional quantities in the gn at 


* 


ay+ wide = 0, 


the differential of U = 0 taken with respect to x,y, and c. It will 
then have the form | 

Mdz-+-Ndy+-Pde = 0; 
whence we derive 


dy a—Tde = de dx = = dy— = dé, ¢ 
and if the integral functions MM, N, are algebraic, or even transcen- 
dental, provided they be not susceptible of becoming infinite by 
some value of c, the coefficient of dc will not disappear but on the 
supposition that P = 0, which thus will give all the particular so- 
lutions of V = 0. 


* 


. 


310 THE INTEGRAL CALCULUS. 


305. When the equation P = 0 contains only c and constant 
quantities, it gives a constant value for c, and of course conducts 
us only to a particular integral. Whence rises only to the first 
degree in the expression of U, it will not enter at all into P, which 
therefore will be composed only of the variables and 7; but in 
this case the equation P = 0, itself satisfies V = 0; for, U = 0 
being of the form Q+cP = 0, the equation V = 0 becomes 
PdQ—QdP = 0. To decide now whether P = 0 is a particular 
solution, or only a particular integral, we must eliminate one of the 
variables x, or y, between U = 0 and P = 0; the result will give 
ce variable in the former case, and constant in the latter. If we 


0 mS 
were to find c = 9’ we must conclude that the equation P = 0 1s 


a factor of U = 0 independently of the constant c, and conse- 
quently is extraneous to the differential equation V = 0. 
306. We shall now apply this theory to the equation 


yda—a«dy = ni/(da?+dy’), 


whose complete integral is y—ex = 2,/( Ic), (299) ; PWS make 
c vary at the same time with « and y, and reduce all the terms to 
the same denominator, we shall obtain 


cda,/(1--e?) —dy,/(1+c°) + § t4/ (1-67) -bnc) gc 0 : 
and putting the coefficient of de equal to zero, it becomes 
t4/(1-+-c?) tne = 0, | 
4, x : ay : 3 
Wie This value of c changes the 
equation y—cxr = n4/(1-+c?) into x?-+-y" = n?, and gives the par~ 
ticular solution obtained in the article referred to. 
All equations of the form y = px-+-P, (299), in which the fore- 
going is comprised, have an analogous particular solution. 
Their complete integral (represented by y = cx+C, C being com- 


posed of c, in the same manner that P is of p, or being the same 
function of it,) gives 


whence we derive c = 


edz = a de =0; 


dC 
and making x-- mm 0, we thence obtain the value of ¢ on 


a 
which the poral solution depends. This particular solution 
made. its appearance in integrating the equation y= = pxt+P; for, 


% 


PHE iNTEGRAL CALCULUS. | * Stl 
in differentiating it, an equation was obtained, composed of the 
two factors . . 


- a = 0,anddp=0, * 
4p Pk 
and the result of the elimination of p between 


dP 
y = px+P, a fait da, = 0, 


a 


would be the same as that of the elimination of c between 


¥ af: aC 
y = cu-+C, and x-- Tae 0, 
The equation y = px+P, P being a function of the differential 
coeflicicient, was first proposed and integrated by CLarraut. 


Let us take, for a further application of the above theory, the 
equation 


* adz--ydy = dy/(x?+-y?—a), 
whose integral is Me 

J (x? y? — a?) = yc, or x*—2cy—c’—a* = 0, 
when the radical is taken away, we find 

zdx—cdy—(y--c) de = 0, 

whence ; , 

3 yte = 0, 
and consequently 

! J (2? +y—a*) = 0; 

ihe particular solution is therefore in this example 
| | x? y?— a? = 0. 


307. One property of particular solutions which presents itself 
easily in this latter example, and which is universal, is that the dzf- 
ferential equation may be so prepared that the particular solution 
shall become a factor of it. In fact, if we put 


4/(a2-by? =a?) = w, 


we shall have 


a 
312 THE INTEGRAL VALCULUS. 


ot cdz-+-ydy = udu, 

and vf proposed equation becomes . 
| udu—udy = 0, 

If we had taken “= Beha, the radical would have mani- 


' fested itself in the transformed equation, which would have be- 


come 
f du—2dy,/u = 0; 


and differentiating, we should have obtained 


du—2d*1 wyun t= 0, 


or d3 uf u—2ud*y — oat, = 0, 
an equation which would still be verified by the supposition of 
«== Q. As these transformations may be continued as far as we 
please, it follows that there are methods of preparing all, the suc- 
cessive differentials of the proposed equation, so that the particular 
solution shall satisfy them likewise, which could’ not be unless 
this proposition were true ; for although, when we make the con- 


d ! 
stant c variable, and put — = Q, we have dy = pdz, as well for 
the solution as for the complete integral, yet the value of d’y, given 
by the particular pelutign, becomes ? 

= P de r ae de © as’ 


while it is simply | da* for the complete integral ; it is not even 


the same factor in these two values, generally Rae » sa- 
tisfy : the equation 


Pu /u—2d'yu—dydu = 0, 


is, as we see, verified by the particular pen indapetidenty of 
the differentials of the second order. 

308. We shall conclude this section by Hhivialhe some ap- 
plications of the formula of Crarravr. 

Pros. 1. To find the curve, such, that perpendiculars from two 
given points to the tangent shall contain a rectangle of a given mag- 
nitude. . 

Let the equation of the tangent be 


tae Se 


(y' —y) ple =o) ge, 


fi 


THE INTEGRAL CALCULUS. 313 


Xu 


the co-ordinates y and x corresponding to the points of contact, 
Let the points from which the perpendiculars are drawn be taken 
upon the axis of x at equal distances -+c and —c on different sides 
of the origin. Hence the two perpendiculars are 


pata sO ee aw 0 
J (ip)? 4/(1-Ep?) 


Let the product of these be b?; .°. 
y’ — p*(c? — a") + 2pzy = b*(1-tp’), 


This solved for y, gives 
y = pacia/(b*+-0°'), 
where a? = 6?-+-c2., Hence, if k be an arbitrary constant, the ge- 
neral solution is 
y = kat y (b?-+-a7k"), 
which is the equation of the tangents to the sought curves, 
The particular solution is - 


ay? ba? = 0%, 


which is the equation of an ellipse. 

If the two perpendiculars be supposed to be drawn to opposite 
sides of the axis of x, their product is negative, .°, 6*<0. In 
this case the particular solution becomes 

 0?y? — 6°? = — 07h’, 
which is the equation of the hyperbola. 

This is a well known property of those curves. 

Pros. I. To find the curve such, thata perpendicular from «@ 
given point upon its tangent shall have a constant length. 


The equation of the tangent being represented as before, let the 
given point be the origin, and let the constant length of the tan- 


gent be r, .*. 
PH 
JCP) 
coy = petri (itp’), 
Hence the singular solution is 
ya = -, 
2$' 


p 
] 


Mi 


314 THE INTEGRAL CALCULUS. 


The circle which results from this particular solution, is the 
locus of the intersections of the straight lines which are expressed 
by giving every possible value toc, in the general solution, and 
which are likewise tangents to the circle. 

Pros. III. To find the curve such, that perpendiculars to a given 
right line from two given points upon that line drawn to meet the 
tangent shall include a given rectangle. 

The equation of the tangent being represented as before, let the 
given line be taken as the axis of 2, the origin being at the middle 
point of the intercept between the given points. Let the distances 
of the given points from the origin be +-a and —a. Hence the 
two perpendiculars | 

y+p(a—z); 
y= p(a—a)- 


Let the constant value of the rectangle under these be 0°. 
Hence 


yf ~ p*(a'—a") — 2pya = b°; 
oo Y = pxrts/(b?+a*p’). 
Hence the general solution is 
y = kat ./(?-+ha"), 
k being an arbitrary constant. And the particular solution is 
a?y?+- 522" = ab’, 


which is an ellipse or hyperbola, according as b? is >0or <0, 
that is, according as the Ber PaRaieulare: are on the same or diflexent 
sides of the axis of a. . 

Pros. IV. To find a curve such, that the locus of the wtersection 
of a perpendicular fror a given point with its tangent shall bea | 
right line. 

Let a perpendicular through the given point be drawn to the 
right line which is the supposed locus, and let these be assumed as 
axes of co-ordinates, the distance of the point from the supposed 


locus being a. ‘The equation of the perpendicular to the tangent 
through the given point is 


rte (c—a) = 0. 


The value of y corresponding to x = 0, is therefore “Hence 


THE INTEGRAL CALCULUS. 315 


the intercept of the perpendicular between the given point and 


supposed locus is 
ag 2 2 
© A kara 
P P 


Hence 
ytp(amz) | V(itp’) 
ov (i-+p") eee 
yand x being the co-ordinates of the point of contact. The 
singular solution of which is 
y? = 4ax, 


the equation of a parabola. 


SECTION XII. 


The Integration of Differential Equations of the Second 
Order. 


309. The difficulty of the integration of equations becomes so 
much the greater the higher the order of the differential coefficients 
which they involve, and we only succeed in effecting it in a very 
small number of limited equations. We shall therefore confine our- 
selves to the investigation of the methods of integrating the most 
important forms of equations of the second order. 

310. If «be taken as the independent variable, the most ge- 
neral form for a differential equation of the second order is 


dy d?y 
Sw a aaa) = © 
or, representing the first and second differential coefficients by y', y",_ 


F(t. 4 y's ¥') = 0. 


Let us first consider the methods of integrating differential 
equations of the second order in the five following cases : 


y 


$16 THE INTEGRAL CALCULUS. 


i. Where they contain only the second differential coefficient 
and the independent variable. 


Il Where they contain only the second differential coefficient 
and the dependent variable. 

III. Where they contain the two differential coefficients and 
neither of the variables. 


IV. Where they contain the two differential coefficients and 
the independent variable. 


V. Where they contain the two differential coefficients and the 
dependent variable. 


811. I. In this case the differential equation is of the form 


d? ” 
F (2, 9) = 0 or F(a, y’) = 0; 


whence we derive 


where X is a function of x ; multiplying by dz and integrating we 
find 
Cee ne 


This being an equation of the first order, may be integrated by 
the methods already explaiiied, and its integral will be of the form 
dy 

dz 

or dy = X'dz-++-Cdz, 


€ being an arbitrary constant. This being again integrated, gives 
an equation of the form : 


= X'+4C, 


y = X"4+-Cr+C, 
C’ being another arbitrary constant. 


Thus, let the equation be 


PE k sy) aa” 6 he = = andy 
ta 0 8° dx : 


and by integtating, we have 


dy ann 
ae n+l Ta 


~~ 


MIE INTEGRAL CALCULUS. 317 


ax t'dx 


or dy = a+ Cae. 


This being again integrated, we find 


agnt? 
“(n+-1)(n +2) 
312. II. When the differential equation is of the form 


F(a B80 


let the equation be supposed to be solved for = and therefore 


reduced to the form 


y ==) ——-+- CrtC. 


dy 
5 laa 

where Y is a function of y. Let both be multiplied by 2dy and in- 
tegrated, and we find 


dy? 
dz? = 2fYdy+C, 


C being an arbitraty constant. The integral fYdy may be deter- 


mined by the established rules. By extracting the square root, we 
shall obtain 


= of (C+2/Ydy) ; 
whence we will derive by a new lle 


GL, 


c= (Te Serta 
Let us take, for example, the differential equation 
2d?y+-yda? Zexs. (Fol 'a? 


d’y a 
dete G45 
dy dy 2ydy * 
dx de. a” 
and by integrating . 


318 THE INTEGRAL CALCULUS: 
AS ae 
dz ~ ae $? 


which is an equation of the first order. 
313.° III. When the differential equation does not include either 
of the variables, it may meer o be reduced to an equation of 


the first order by substituting | —— 7, for by which it becomes 


dy 
ae 


dy, 
» (hs) = Q. 


We derive from this equation 


Y being a function of y, and consequently 


du’ 
aisey fied) od Caaidon’, « eyiggh(l)s 


On the other hand, the equation = = 4, gives 


y = fy de ; 


and by substituting in this equation the value of dx, above obtain- 
ed, we shall have 


y JPEN 4 ONG ae! 


Then it only remains to eliminate y' between the two equations (1) 
and (2), the result, including two arbitrary constants, will be the 
integral sought. It will likewise be complete, since it involves two 
arbitrary constants ; and we have seen in art (69), that this is the 
greatest number which the integral of any equation of the second 
degree can possibly contain. The elimination of y’ cannot be ef- 
fected unless we shall have previously effected the integrations in- 
dicated; but by means of quadratures we shall be able to construct 
the curve which may be sought for. 

Let us take, for example, the equation 


(dx?4-dy?) ? 
dxd?y 


By putting y/dx for dy, dy'dx for d’y, we shall change this equation 
into 


THE INTEGRAL CALCULUS. 319° 


dm whiciews deduce 


ady ay dy 
dx = J 33 dy=y si OR a 
(1+y'*)? (i-ry”) ? 
The integration will give 
ay a 
g = C+ —— 2, y = C -—_—- | ; 
SAY I 
eliminating y’, we shall get 
(z—C)+(y—C) = a’. 

The proposed differential equation is nothing more than the ge- 
neral expression of the radius of-eurvature, made equal to a con- 
stant quantity a (133); and as we ought to expect, the integral is 
the equation to a circle of which this constant quantity is the radius. 

314. IV. A differential equation of the second order of the form 

wege dy 
> a ct) = 0 or f(x, y', y”) = 0, 
is reduced immediately to one of the first order by substituting 


— for y’. The equation, therefore, assumes the form 


d f 
ows H) =o 
which is an equation of the first order between 7 and x. This be- 
ing integrated by the established rules, gives an equation of the form 
f(a, y',¢) = 9, ¢ ae an arbitrary constant. 
Again, Papanatiag 2 7 4 for y', this becomes a differential equation 


of the first order between y and x. 


This being integrated, as be- 
fore, gives an equation of the form 


F(%, y ©, ¢) =0, 
ec and’ being the two arbitrary constants. 


le?+-dy?)? 2 
Thus, let Os na = lees Hence, 


320 THE INTEGRAL CALCULUS, 


3 
* (1--y? _ aoe a? 
a ae 
N oy ° dy’ e 
For y' substitute Ta and we obtain 
dy 2edze» 3 y __ #*-ac 
_ Smit ae 


(yt ty? 
and consequently, 


x?-+-ac \ _ op (x*-+ac)dx 


Y= Fram etany Sy yaim ebay 

The integral of which is obtained by the rules already establish- 
ed. ‘This is the equation of the elastic curve. 

315. In general, the equation f(z, y', c) = 0, may be integrated 
by three different methods, which may be chosen according as they 
may severally be found best suited to the circumstances of the pro- 


posed equation. 
First. If the equation admit of being vee for y', it may be re- 


duced to the form 


d 
“P=; 1. y = fXde. 


Secondly. If it admit of being solved for x, it may be reduced 
to the form 


a= Fy). 
But ydx = dy; ©. y =fy'dx = yx—fady. Hence 
y = yx — fly) dy. 
The latter integral being determined, y may be eliminated by 


means of this equation and x = f(y’), and the result will give the 


integral sought. 

Thirdly. If the equation do not’admit of solution for x or y, it 
will necessary, by a transformation, to express x and y’ as functions 
of a third variable z ; let these functions be z, 2’, so that 


em2z,y =235 4 = fide = fe'dz, 


which is the integral sought. 
- 316. V. If the equation have the form 


 ; Se 


THE INTEGRAL CALCULUS. 321° 


. dy d2y 
f(» Phi sr) = 0 or flys vy") = 0, 


it may be reduced to one of the first order ; thus, 
dy = ydz,ydz = dy; 
egy aan 1 Yay 
. ydy = y"dy, and y' = dah 
Let the proposed form be expressed thus, 
gy =f¥, ys o. Yay =SY's y)ay: 
This being a differential equation of the first order between y’ and 
y, may be integrated by the usual metliods, and its integral will 
have the form F(y. y', c) = 0, ¢ being the arbitrary constant. The 
integration of this presents two cases : 
First. Where the variables may be separated, and therefore the 
equation may be reduced to either of the forms 


y= Y,y= YX, 


Y and Y’ being functions of y and y respectively. 
In the first case the integration is effected by reducing it to the 
form 


OL alee ORY 64 
de=s a= Jody. 


_ dn the second case, since dy = y‘dz ; 
*ydxe = dY, 


and consequently 


dx = 3 Saket =f. 


j Eliminating y’ by this and the equation y = ¥’, the resulting equa- 
tion will be the integral sought. 
Let there be, for example, the differential anti 


yyy +a) = y(ity”). 
Since y’dy = y'dy/, this is reduced to 
dy'(yy' ra) = dy (1-+y"). 
This being integrated by Clairaut’s formula (306), gives 
y = ay Pe (1-Fy”), | 
27 


329 THE INTEGRAL CALCULUS. 


pee dy AY f #y ’ eo 
a fF = alby) tly ty (ly). 


Eliminating 7/ by these equations, the integral may be found. 

Secondly. If the variables y and y cannot be separated, a 
transformation may be effected by expressing y and y' as functions, 
z, 2’, of another variable z. Since y'dx = dy; .. 2dx = dz, 
and the integration may be effected as in the last case of (315). 

317 There are some remarkable cases in which differential equa- 
tions of the second order, where they include both the varia‘les, - 
may be integrated without much difficulty. We shall consider the 
_ three following cases : 


d?y : dy A oti 
d?y dy wil aaa 
fs MAX EX y AX" = 0, 


(where X, X’, X”, are any functions of 2.) 
{f1. When the equation is homogeneous with respect to the va- 
riables and their differentials. 


d 
318.1. Let y= ele, ) °, “ = wesude, and 


Making these substitutions, we find 
oy (u?-++- Xu-+ X’) = 0, 


since the common factor e/““* disappears. This, being a differen- 
tial equation of the first order, is integrated by the methods esta- 
blished in Section X. ; 
319. II. This equation may be reduced to the preceding by any 
transformation that will remove the term X”. For this purpose, 
let y= tz. Hence 


dy dz , dedt dt 
dx? du? dadx ©” dxt- 


Making these substitutions, we find 


ae 


THE INTEGRAL pAbavEte, 323 
dz dzdt dz dt ALT 
(+22 _ coi of (+ 2 )+X'te+X = 0. 


Let the variable z be limited by the condition 
aT ge ley 2 OR or ah be (1). 


Hence the transformed aerate, after dividing by z, becomes 
dt 2dz\ , X” 
PDE, oc6 9 -) 


; The first (1) of these equations may be integrated by the pre- 
ceding article, and thence an equation found of the form FYz, x) 


; 2s 3 : 
=: 0. By this process i will become known as a function of , 
x 


and thus the equation (2) will become integrable. The process in 
general may be conducted thus. Let 


= eJ/Xde, +) Xdx we 
. v 
by this substitution, oe (2) Bocas 
hate Qdz\ _ X"dx 


But since 


the equation may be reduced to the form 


a t vat dl (v2") +X" vzdx = 0. 


Integrating it under this form, we find 


_ p2°-+-fX'ozdx = 0; i 


rl & = oP es = 0. 


But since a J. a“ yt2 (SF X"vedx )= 0. 


z 
In this, v isa given function of x by the equation v = e/X@, and 
zisafunction of « by (i). Hence this last. equation is the in- 
tegral sought. It will obviously include two arbitrary constants. 


324 THE INTEGRAL CALCULUS. 


320. Ill. If the equation be homogeneous with respect to y, 2; 
dy, d*y, and dz, let 


di , a2 2 
y = un, aes, 


at, 4, and z, being considered as new variables. By this substi- 
tution, every term of the equation will have the same.power of x : 
as a factor, which being removed by division, the equation assumes 
the form 


F(y, 2, u) = 0, orz = f(y’, u). 
By differentiating y = uz, we find 
dy = udx-+-adu; .. y'dx = udz-+adu ; 


deo edu 
Se y' — 
gg ne 2 d?y 
Also, since Te a? oe ics zdx, 
and consequently 
dz. dy 
. ady == zdx, or — = as 
xu < 


Hence 


du 

as Sa a | eee ee a See ed = 
Zz y—u’ * 4 y! =u SIs) ee 

This being an equation of the first order between y’ and u, may be 

integrated by the rules for integrating such equations. The inte- 
gration will give y’ = F(w). Hence 


a du 1 
ah Fy a Sipe 


Eliminating u by this and y = uz, we obtain the integral of the 
proposed equation. It is obvious that this integral will include two 
arbitrary constants introduced by the two integrations effected prior 
to the elimination. 

Thus, let xd’y = dydx; .. z = y', and hence, 

dy’ __—s du 


_—_—— or 


, 


y jaa 


Bh pane 
ot : Yy? = f(udy+y'du) = wuts - 


But, also 


THE INTEGRAL CALCULUS. 325 


dx dy’ , 
— = 5 we = ay’. 
a y . 
Eliminating y' by these equations, the result is «?—2axu = C. 
Eliminating «, we obtain the integral 

x? 2ay = C. 


321. We shall conclude by subjoining a few examples, to exer- 


cise the learner in the practical ayer of the principles laid 
down in this Section. 


mx. 1. eye ey hy. Mal where 


dz?"dz2_ a b 
= 4/(dx?+dy’) is constant ; 
. ds d*s = dzd’a+dyd’y = 


i i ly'dx mee a 
and since a =— “ssyt =~, where y =<, 
2, 
ody = (toa) 3 = ds*d7y 


~dx3’ 


and consequently 
ie "Vidi than "Dy k's Seek Oe @ 
dy’ = — cos 7; cary = sin pr 6; 


hence, by substitution, 


6 x 
eicaanr 008 preete, 


e and ¢’ being arbitrary constants. 

Ex. 2. A body moves uniformly along a given right line, and 
another moves uniformly in pursurt of it, to find the path of the 
laiter. 

Let the given right line be the axis of x, and let x, y, be the co- 
ordinates of the place of the pursuer, and let c be the exponent of 
the ratio of the velocities of the two bodies. The pursuer may be 
considered at each instant as moving in the tangent to the curve of 
pursuit, and the tangent itself as continually passing through the 
pursued body. 


The distance of the point where the tangent meets the axis of « 
from the origin is 


326 THE INTEGRAL CALCULUS. 
wdy—ydar 
dy. * 

Now if s be the are of the curve of pursuit measured from the — 


point where the tangent is perpendicular to the axis of x, in which 
position we may assume the axis of y, we have 


ady —ydx iy 
dy 
Differentiating this equation, considering y as the sid at 
- variable, we find 
d?x aGey 
oS (dye da?) oy” 
i at cdy da’ 
t er OS 
NE ge dye Fas eee 
we ay / (12'S = (ay); 
where a is an arbitrary constant. Hence 
on ccyots l 
— B(e-b1) | Ba(om 1)ye-1 
This is the simplest case of curves of pursuit. 
Ex. 3. Let a?d?y = ydx*; 


+C. 


dydx? 
*, dyd’y = 4 a, and 


| dy? _ yte 
dx2 a? ? 
ad +¥ (ye 
.dz= Cry and 7 = al ; Ae 


or, expressing y in terms of x, we find 


y= Cee +C’e * 
Ex. 4. Let d’y,/(ay) —da? = 


{8 


oo 


dy 4 Pape 
fan 5 /ay-+C, changing C into Fa! we shall deduce from 
thence 
yr diy? 4 j Odx eee 


a tin re (4/yC) 5 Va V(etVy) ° 


ae . 


THE INTEGRAL CALCULUS. 327 


94 
and 2 2 YE} (yy = 20) yeetw) § +E 
Ex. 5. Let ad bya Cee Wer) = 0. Ret = = y; 
oy 2e9e pie 
9 a ia ie at The equation, therefore, becomes 


aL +(14y? = 0; 
ady 
(1-4y8 
and since dy = ydr; .«. dy = ey a 
(i+y’?)? 


Integrating these, we obtain 


ee dx —— 


, 


»v 


BE ye Cty? 


a 


Eliminating 7, we find 


(A—2)'-+-(B—y)? = 
This example proves that the circle is the only curve of which the 
radius of curvature is constant, 


Ex. 6. To find the curve of which the radius of curvature varies. 
inversely as the abscissa. 


By (133) 
(dxttdy)® (ty?) 
ar, ~~ dxd?y See ie 
Since ¥ varies inversely as «, let 
G4 
‘im en? 


a being constant. Hence the equation to be integrated is 
ay’ +-22( 12)? = 0. 


This has already been integrated in (314), and the wo is the 
equation of the elastic curve. 


325 THE INTEGRAL CALCULUS. 


Ex. 7. To find the curve in which the radius of curvature is 
given function of the abscissa. 
In this case 


This problem solves all the inverse problems relating to the radius 
of curvature. é 


Ex. 8. Let, y’-+2-—2 4%. = 0, 
cae ST aaah Loa 
Comparing this with the formula B ea in article (319), we find 
Rae, oy ae oe 
Rg Tee ee 


Hence the equation (1) art. (319), becomes 


d2z-1 d: 
ap SE, 


dz? ‘adx x? 


which by putting z = e/*4*, gives 


du l 
alt +2-3)= 


‘ Aig ‘ i 
This equation is rendered homogeneous by making u = ra. the 


variables are there separated by putting « = sw. Hence 


du’ aed j ee peas é 
oy, ee ‘o'r ’ 
w a s s—1 
neglecting the constant. Substituting for uw’ and s, their sae we 
find , 
plas os a x? — I] 7m I 

aS rae Slide 0) a ~ 

Also v = e/Xde =e = x. Hence we find 


fxX"vzdx = fadx = idee 
and, therefore, — 


x2 — J a b dz: 
fs ail ee ? 


which is integrated by the rules for as differentials. 


dl 


yi 
4 L 
’ @ 
' 
} 
“4 ' 
mis 
pee 
re x 
. Q Pee 
=o | f 
& 
“ 
¥ 5 | 
4 = 
nes ner ee 
7 
* 


a! Df -0-0-0-0-0 0-0-0090 0-90-90 9-9 9 9 
20 4-4 B10 909-98 


RY ) 
| r 


ge ae 


Plate 2. 


a8 
CK 


2. P d 


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a 
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7 
we i ¥ * 
4 4 a 


Fy e 


pore Le achat ferme me STEEN Sep RNG FN ET 


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